foundations of computational geometric...

Foundations of Computational Geometric Mechanics

Thesis byMelvin Leok

In Partial Fulfillment of the Requirementsfor the Degree of

Doctor of Philosophy

California Institute of TechnologyPasadena, California

2004

(Defended May 6, 2004)

iii

c© 2004

Melvin Leok

All Rights Reserved

v

“Thought is only a flash between two long nights,

but this flash is everything”

Henri Poincare, 1854–1912.

vii

Preface

This thesis was submitted at the California Institute of Technology on April 19, 2004, in partial

fulfillment of the requirements for the degree of Doctor of Philosophy in Control and Dynamical

Systems with a minor in Applied and Computational Mathematics. The thesis was defended on

May 6, 2003, in Pasadena, CA, and was approved by the following thesis committee:

• Thomas Y. Hou, Applied and Computational Mathematics,

California Institute of Technology,

• Jerrold E. Marsden (Chair), Control and Dynamical Systems,


• Richard M. Murray, Control and Dynamical Systems, and Mechanical Engineering,


• Michael Ortiz, Aeronautics, and Mechanical Engineering,


• Alan D. Weinstein, Mathematics,

University of California, Berkeley.

This thesis draws upon the following papers that have or will be submitted for publication, but an

effort has been made to clarify the manner in which the research relates to each other by incorporating

extended introductions and conclusions within each chapter, and through the use of examples that

draw upon the various chapters.

• S.M. Jalnapurkar, M. Leok, J.E. Marsden, and M. West, Discrete Routh Reduction, J. FoCM,

submitted.

• M. Desbrun, A.N. Hirani, M. Leok, and J.E. Marsden, Discrete Exterior Calculus, in prepara-

tion.

• M. Desbrun, A.N. Hirani, M. Leok, and J.E. Marsden, Discrete Poincare Lemma, Appl. Nu-

mer. Math., submitted.

viii

• M. Leok, J.E. Marsden, A.D. Weinstein, A Discrete Theory of Connections on Principal Bun-

dles, in preparation.

• M. Leok, Generalized Galerkin Variational Integrators, in preparation.

A review article based on preliminary versions of the work on discrete exterior calculus, discrete

Poincare lemma, and discrete connections on principal bundles, received the SIAM Student Paper

Prize, and the Leslie Fox Prize in Numerical Analysis (second prize), both in 2003.

ix

Acknowledgements

It has been an honor and a privilege to work with my advisor, Jerrold Marsden, and it is an

opportunity for which I owe him a tremendous debt of gratitude. His support, guidance, and

invaluable insight has steered my mathematical education since my early years as an undergraduate,

and sparked my interest in geometric mechanics, discrete geometry, and its relation to computational

methods.

I would like to thank my collaborators Mathieu Desbrun, Anil Hirani, Sameer Jalnapurkar, Alan

Weinstein, and Matthew West. Aside from contributing markedly to my education, much of this

work would not have been possible without discussions, insights, and much needed motivation from

them.

I am grateful to the members of my thesis committee —Thomas Hou, Jerrold Marsden (Chair),

Richard Murray, Michael Ortiz, and Alan Weinstein. I would also like to thank my references,

Mathieu Desbrun, Darryl Holm, Arieh Iserles, Lim Hock, Thomas Hou, and Jerrold Marsden, for

the countless letters they have written in my support.

During my travels, I have experienced the warm hospitality of John Butcher, Michael Dellnitz,

Darryl Holm, Arieh Iserles, Hans Munthe-Kaas, Marcel Oliver, and Brynjulf Owren. They have

each been supportive in their own way, and helped shaped my research interests, for which I am

thankful.

This work has been supported in part by a Caltech Poincare Fellowship (Betty and Gordon Moore

Fellowship), an International Fellowship from the Agency for Science, Technology, and Research,

Singapore, a Josephine de Karman Fellowship, a Tan Kah Kee Foundation Postgraduate Scholarship,

a Tau Beta Pi Fellowship, and a Sigma Xi Grant-in-Aid of Research. Additional travel support was

provided by an ICIAM travel grant from SIAM, a London Mathematical Society bursary, and a

SIAM student travel award.

I would like to acknowledge my fellow graduate students, Harish Bhat, Domitilla del Vecchio,

Melvin Flores, Xin Liu, Shreesh Mysore, Antonis Papachristodoulo, and Stephen Prajna, for making

my graduate experience that much more rich and pleasant.

My thanks to the staff at Caltech, including Charmaine Boyd, Cherie Galvez, Betty-Sue Herrala,

x

Arpy Hindoyan, Wendy McKay, and Michelle Vine, for helping me navigate the intricacies of life

at Caltech, and beyond. Many thanks also to the current and former staff at International Student

Programs, including Parandeh Kia, Tara Tram, and Jim Endrizzi.

The typesetting of this thesis and the figures herein would not have been possible without the

LATEX expertise of Wendy McKay and Ross Moore.

A special thanks to my parents, Belinda and James, for their unconditional love and support,

and for the freedom to pursue my interests. I dedicate this thesis to Lina, who has made my life so

much better in countless ways.

xi

Foundations of Computational Geometric Mechanics

by

Melvin Leok

In Partial Fulfillment of the

Requirements for the Degree of

Doctor of Philosophy

Abstract

Geometric mechanics involves the study of Lagrangian and Hamiltonian mechanics using geometric

and symmetry techniques. Computational algorithms obtained from a discrete Hamilton’s principle

yield a discrete analogue of Lagrangian mechanics, and they exhibit excellent structure-preserving

properties that can be ascribed to their variational derivation.

We construct discrete analogues of the geometric and symmetry methods underlying geometric

mechanics to enable the systematic development of computational geometric mechanics. In par-

ticular, we develop discrete theories of reduction by symmetry, exterior calculus, connections on

principal bundles, as well as generalizations of variational integrators.

Discrete Routh reduction is developed for abelian symmetries, and extended to systems with

constraints and forcing. Variational Runge–Kutta discretizations are considered in detail, includ-

ing the extent to which symmetry reduction and discretization commute. In addition, we obtain

the Reduced Symplectic Runge–Kutta algorithm, which is a discrete analogue of cotangent bundle

reduction.

Discrete exterior calculus is modeled on a primal simplicial complex, and a dual circumcentric

cell complex. Discrete notions of differential forms, exterior derivatives, Hodge stars, codifferentials,

sharps, flats, wedge products, contraction, Lie derivative, and the Poincare lemma are introduced,

and their discrete properties are analyzed. In examples such as harmonic maps and electromag-

netism, discretizations arising from discrete exterior calculus commute with taking variations in

Hamilton’s principle, which implies that directly discretizing these equations yield numerical schemes

that have the structure-preserving properties associated with variational schemes.

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Discrete connections on principal bundles are obtained by introducing the discrete Atiyah se-

quence, and considering splittings of the sequence. Equivalent representations of a discrete connec-

tion are considered, and an extension of the pair groupoid composition that takes into account the

principal bundle structure is introduced. Discrete connections provide an intrinsic coordinatization

of the reduced discrete space, and the necessary discrete geometry to develop more general discrete

symmetry reduction techniques.

Generalized Galerkin variational integrators are obtained by discretizing the action integral

through appropriate choices of finite-dimensional function space and numerical quadrature. Explicit

expressions for Lie group, higher-order Euler–Poincare, higher-order symplectic-energy-momentum,

and pseudospectral variational integrators are presented, and extensions such as spatio-temporally

adaptive and multiscale variational integrators are briefly described.

xiii

Contents

1 Introduction 1

2 Discrete Routh Reduction 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Continuous Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Discrete Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Discrete Variational Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Discrete Mechanical Systems with Symmetry . . . . . . . . . . . . . . . . . . 17

2.3.3 Discrete Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.4 Preservation of the Reduced Discrete Symplectic Form . . . . . . . . . . . . . 27

2.3.5 Relating Discrete and Continuous Reduction . . . . . . . . . . . . . . . . . . 34

2.4 Relating the DEL Equations to Symplectic Runge–Kutta Algorithms . . . . . . . . . 35

2.5 Reduction of the Symplectic Runge–Kutta Algorithm . . . . . . . . . . . . . . . . . 38

2.6 Putting Everything Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7 Links with the Classical Routh Equations . . . . . . . . . . . . . . . . . . . . . . . . 43

2.8 Forced and Constrained Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.8.1 Constrained Coordinate Formalism . . . . . . . . . . . . . . . . . . . . . . . . 46

2.8.2 Routh Reduction with Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.8.3 Routh Reduction with Constraints and Forcing . . . . . . . . . . . . . . . . . 56

2.9 Example: J2 Satellite Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.9.1 Configuration Space and Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 57

2.9.2 Symmetry Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.9.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.9.4 Discrete Lagrangian System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.9.5 Example Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.9.6 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.10 Example: Double Spherical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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2.10.1 Configuration Space and Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 63

2.10.2 Symmetry Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.10.3 Example Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.10.4 Computational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.11 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3 Discrete Exterior Calculus 73

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2 History and Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3 Primal Simplicial Complex and Dual Cell Complex . . . . . . . . . . . . . . . . . . . 78

3.4 Local and Global Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 Differential Forms and Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . 87

3.6 Hodge Star and Codifferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.7 Maps between 1-Forms and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . 92

3.8 Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.9 Divergence and Laplace–Beltrami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.10 Contraction and Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.11 Discrete Poincare Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.12 Discrete Variational Mechanics and DEC . . . . . . . . . . . . . . . . . . . . . . . . 121

3.13 Extensions to Dynamic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.13.1 Groupoid Interpretation of Discrete Variational Mechanics . . . . . . . . . . . 129

3.13.2 Discrete Diffeomorphisms and Discrete Flows . . . . . . . . . . . . . . . . . . 131

3.13.3 Push-Forward and Pull-Back of Discrete Vector Fields and Discrete Forms . . 134

3.14 Remeshing Cochains and Multigrid Extensions . . . . . . . . . . . . . . . . . . . . . 136


4 Discrete Connections on Principal Bundles 139

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.2 General Theory of Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.3 Connections and Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.4 Discrete Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.4.1 Horizontal and Vertical Subspaces of Q×Q . . . . . . . . . . . . . . . . . . . 151

4.4.2 Discrete Atiyah Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.4.3 Equivalent Representations of a Discrete Connection . . . . . . . . . . . . . . 155

4.4.4 Discrete Connection 1-Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.4.5 Discrete Horizontal Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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4.4.6 Splitting of the Discrete Atiyah Sequence (Connection 1-Form) . . . . . . . . 165

4.4.7 Splitting of the Discrete Atiyah Sequence (Horizontal Lift) . . . . . . . . . . 167

4.4.8 Isomorphism between (Q×Q)/G and (S × S)⊕ G . . . . . . . . . . . . . . . 168

4.4.9 Discrete Horizontal and Vertical Subspaces Revisited . . . . . . . . . . . . . . 170

4.5 Geometric Structures Derived from the Discrete Connection . . . . . . . . . . . . . . 172

4.5.1 Extending the Pair Groupoid Composition . . . . . . . . . . . . . . . . . . . 173

4.5.2 Continuous Connections from Discrete Connections . . . . . . . . . . . . . . . 176

4.5.3 Connection-Like Structures on Higher-Order Tangent Bundles . . . . . . . . . 176

4.6 Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.6.1 Exact Discrete Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.6.2 Order of Approximation of a Connection . . . . . . . . . . . . . . . . . . . . . 180

4.6.3 Discrete Connections from Exponentiated Continuous Connections . . . . . . 181

4.6.4 Discrete Mechanical Connections and Discrete Lagrangians . . . . . . . . . . 182

4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

4.7.1 Discrete Lagrangian Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.7.2 Geometric Control Theory and Formations . . . . . . . . . . . . . . . . . . . 186

4.7.3 Discrete Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . . 188


5 Generalized Variational Integrators 193

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

5.1.1 Standard Formulation of Discrete Mechanics . . . . . . . . . . . . . . . . . . 194

5.1.2 Multisymplectic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

5.1.3 Multisymplectic Variational Integrator . . . . . . . . . . . . . . . . . . . . . . 196

5.2 Generalized Galerkin Variational Integrators . . . . . . . . . . . . . . . . . . . . . . . 197

5.2.1 Special Cases of Generalized Galerkin Variational Integrators . . . . . . . . . 198

5.3 Lie Group Variational Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

5.3.1 Galerkin Variational Integrators . . . . . . . . . . . . . . . . . . . . . . . . . 203

5.3.2 Natural Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5.4 Higher-Order Discrete Euler–Poincare Equations . . . . . . . . . . . . . . . . . . . . 207

5.4.1 Reduced Discrete Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.4.2 Discrete Euler–Poincare Equations . . . . . . . . . . . . . . . . . . . . . . . . 209

5.5 Higher-Order Symplectic-Energy-Momentum Variational Integrators . . . . . . . . . 210

5.6 Spatio-Temporally Adaptive Variational Integrators . . . . . . . . . . . . . . . . . . 212

5.7 Multiscale Variational Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

5.7.1 Multiscale Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

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5.7.2 Multiscale Variational Integrator for the Planar Pendulum with a Stiff Spring 216

5.7.3 Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.8 Pseudospectral Variational Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . 222

5.8.1 Variational Derivation of the Schrodinger Equation . . . . . . . . . . . . . . . 222

5.8.2 Pseudospectral Variational Integrator for the Schrodinger Equation . . . . . . 223


6 Conclusions 231

A Review of Homological Algebra 233

B Geometry of the Special Euclidean Group 237

C Analysis of Multiscale Finite Elements in One Dimension 239

Bibliography 245

Index 257

Vita 265

xvii

List of Figures

2.1 Complete commutative cube. Dashed arrows represent discretization from the con-

tinuous systems on the left face to the discrete systems on the right face. Vertical

arrows represent reduction from the full systems on the top face to the reduced sys-

tems on the bottom face. Front and back faces represent Lagrangian and Hamiltonian

viewpoints, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 Unreduced (left) and reduced (right) views of an inclined elliptic trajectory for the

continuous time system with J2 = 0 (spherical Earth). . . . . . . . . . . . . . . . . 60

2.3 Unreduced (left) and reduced (right) views of an inclined elliptic trajectory for the

continuous time system with J2 = 0.05. Observe that the non-spherical terms intro-

duce precession of the near-elliptic orbit in the symmetry direction. . . . . . . . . . 61

2.4 Unreduced (left) and reduced (right) views of an inclined trajectory of the discrete

system with step-size h = 0.3 and J2 = 0. The initial condition is the same as that

used in Figure 2.2. The numerically introduced precession means that the unreduced

picture looks similar to that of Figure 2.3 with non-zero J2, whereas, by considering

the reduced picture we can see the correct resemblance to the J2 = 0 case of Figure

2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.5 Unreduced (left) and reduced (right) views of an inclined trajectory of the discrete

system with step-size h = 0.3 and J2 = 0.05. The initial condition is the same as that

used in Figure 2.3. The unreduced picture is similar to that of both Figures 2.3 and

2.4. By considering the reduced picture, we obtain the correct resemblance to Figure

2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.6 Double spherical pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.7 Time evolution of r1, r2, ϕ, and the trajectory of m2, relative to m1, using RSPRK. 69

2.8 Relative energy drift (E−E0)/E0 using RSPRK (left) compared to the relative energy

drift in a non-symplectic RK (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.9 Relative error in r1, r2, ϕ, and E between RSPRK and SPRK. . . . . . . . . . . . . 70

3.1 Primal, and dual meshes, as chains in the first circumcentric subdivision. . . . . . . 80

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3.2 Orienting the dual of a cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.3 Examples of chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.4 Discrete vector fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.5 Terms in the wedge product of two discrete 1-forms. . . . . . . . . . . . . . . . . . . 94

3.6 Stencils arising in the double summation for the two triple wedge products. . . . . . 98

3.7 Associativity for closed forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.8 Divergence of a discrete dual vector field. . . . . . . . . . . . . . . . . . . . . . . . . 101

3.9 Laplace–Beltrami of a discrete function. . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.10 Extrusion of 1-simplex by a discrete vector field. . . . . . . . . . . . . . . . . . . . . 104

3.11 Flow of a 1-simplex by a discrete vector field. . . . . . . . . . . . . . . . . . . . . . . 107

3.12 The cone of a simplex is, in general, not expressible as a chain. . . . . . . . . . . . . 108

3.13 Triangulation of a three-dimensional cone. . . . . . . . . . . . . . . . . . . . . . . . . 111

3.14 Trivially star-shaped complex (left); Logically star-shaped complex (right). . . . . . 112

3.15 One-ring cone augmentation of a complex in two dimensions. . . . . . . . . . . . . . 115

3.16 Logically star-shaped complex augmented by cone. . . . . . . . . . . . . . . . . . . . 115

3.17 One-ring cone augmentation of a complex in three dimensions. . . . . . . . . . . . . 117

3.18 Regular tiling of R3 that admits a generalized cone operator. . . . . . . . . . . . . . 118

3.19 Spiral enumeration of Sn−2, n = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.20 Counter-example for the discrete Poincare lemma for a non-contractible complex. . . 120

3.21 Boundary of a dual cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.22 Prismal cell complex decomposition of space-time. . . . . . . . . . . . . . . . . . . . 126

3.23 Groupoid composition and inverses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.24 Spatial and material representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.1 Reorientation of a falling cat at zero angular momentum. . . . . . . . . . . . . . . . 141

4.2 A parallel transport of a vector around a spherical triangle produces a phase shift. . 142

4.3 Geometric phase of the Foucault pendulum. . . . . . . . . . . . . . . . . . . . . . . . 142

4.4 True Polar Wander. Red axis corresponds to the original rotational axis, and the gold

axis corresponds to the instantaneous rotational axis. . . . . . . . . . . . . . . . . . . 143

4.5 Geometry of rigid-body phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.6 Rigid body with internal rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.7 Geometry of a fiber bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.8 Geometric phase and connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.9 Intrinsic representation of the tangent bundle. . . . . . . . . . . . . . . . . . . . . . . 151

4.10 Application of discrete connections to control. . . . . . . . . . . . . . . . . . . . . . . 187

4.11 Discrete curvature as a discrete dual 2-form. . . . . . . . . . . . . . . . . . . . . . . . 190

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5.1 Polynomial interpolation used in higher-order Galerkin variational integrators. . . . 198

5.2 Linear and nonlinear approximation of a characteristic function. . . . . . . . . . . . 213

5.3 Mapping of the base space from the computational to the physical domain. . . . . . 214

5.4 Factoring the discrete section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

5.5 Comparison of the multiscale shape function and the exact solution for the elliptic

problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

5.6 Planar pendulum with a stiff spring . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

5.7 Comparison of the multiscale shape function and the exact solution for the planar

pendulum with a stiff spring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

xxi

List of Tables

3.1 Determining the induced orientation of a dual cell. . . . . . . . . . . . . . . . . . . . 82

3.2 Primal simplices, dual cells, and support volumes in two dimensions. . . . . . . . . . 84

3.3 Primal simplices, dual cells, and support volumes in three dimensions. . . . . . . . . 85

3.4 Causality sign of 2-cells in a (2 + 1)-space-time. . . . . . . . . . . . . . . . . . . . . . 127

1

Chapter 1

Introduction

Geometric mechanics (see, for example, Abraham and Marsden [1978]; Arnold [1989]; Marsden and

Ratiu [1999]) has motivated the development of new and innovative numerical schemes (see, for

example, Kane et al. [1999, 2000]; Marsden and West [2001]; Lew et al. [2003, 2004]) that inherit

many of the desirable conservation properties of the original continuous problem. It is the goal of

computational geometric mechanics to more directly adopt the approach of geometric mechanics in

the construction of computational algorithms, as well as the systematic analysis of their numerical

conservation properties. This is part of the broader subject of structure-preserving integrators,

and good references for this field include Sanz-Serna and Calvo [1994], Hairer et al. [2002], and

Leimkuhler and Reich [2004].

In simulating dynamical systems for a long time, it is often desirable to preserve as many of the

physical invariants as possible, since this typically results in a more qualitatively accurate simula-

tion. As an example, symplectic methods have become popular in simulating the solar system, and

molecular dynamics, where long-time behavior is of paramount importance. While conditions exist

on the coefficients of a Runge–Kutta scheme that ensure that the scheme is symplectic, directly

constructing a symplectic method using this approach can be difficult. An alternative approach is

to discretize Lagrangian mechanics by considering a discrete Hamilton’s principle. The resulting

variational integrators (see, for example, Marsden and West [2001]) have the desirable property that

they are symplectic and momentum preserving. Forcing and dissipation can also be addressed by

considering the discrete Lagrange–d’Alembert principle instead. In either the forced or conservative

case, variational integrators exhibit excellent longtime energy behavior that cannot be understood

from their local error properties alone. Such discrete conservation laws typically impart longtime

numerical stability to computations, since the structure preserving algorithm exactly conserves a

discrete quantity that is always close to the continuous quantity of interest. The longtime stability

of variational integrators has recently been analyzed using the techniques of Γ-convergence in Muller

2

and Ortiz [2004], which yield insights into the weak∗ convergence of discrete trajectories, and the

convergence of the Fourier transformation of the discrete trajectories.

Classical field theories like electromagnetism and general relativity have a rich underlying geome-

try, and understanding the role of gauge symmetries in these problems is important for distinguishing

between the physically relevant dynamics and the nonphysical gauge modes. One method of elimi-

nating the gauge symmetry is by the process of reduction, where the dynamics is restricted to the

constant momentum surface that the flow is on, and the symmetry is further quotiented out. This

results in a reduced system of equations that evolve on a lower-dimensional space referred to as

the shape space. The shape space is the natural setting for studying relative equilibria, such as

rigid-bodies in uniform rotation or translation. Insight is obtained by computing on the shape space

that would otherwise remain obscured at the unreduced level.

Staggered mesh schemes like the Yee scheme in computational electromagnetism, and the leapfrog

scheme in ordinary differential equations, have good structure-preservation properties due to dis-

cretizations that are compatible with the underlying geometry. These geometric relationships are

obscured by the use of vector calculus, but when the equations are reformulated in the language

of differential forms and exterior calculus, the primal-dual relation reflected in the use of staggered

meshes naturally arises. While it is understood how to obtain compatible discretizations for stag-

gered meshes that are logically rectangular, unstructured meshes pose a greater challenge.

In practice, much of the understanding of complex dynamical systems is derived from numerical

simulations which, because of their complexity, are typically not fully resolved. Therefore, the

behavior of numerical algorithms and discrete geometry for finite discretizations is important. The

interaction between numerical methods and the dynamical systems to which they are applied can

be non-trivial, and this naturally leads to the question of how to construct canonical discretizations

that preserve, at a discrete level, the important properties of the continuous system. Computational

geometric mechanics aims, in part, to address this and other questions, but understanding the global

behavior of integrators, and their underlying geometry, we are hindered by the absence of the discrete

analogues of mathematical tools that geometric mechanics has come to rely on. These include the

language of differential forms and exterior calculus, as well as geometric structure encoded in the

form of connections on principal bundles. In recent years, there has been increasing interest in the

study of numerical schemes as dynamical systems in their own right, and if we are to repeat the

success of geometric mechanics in elucidating the underlying geometry of such numerical methods,

it is imperative that we develop more of the relevant mathematical infrastructure.

In particular, there has been interest in developing a theory of symmetry reduction at the discrete

level, including work on the Discrete Euler–Poincare equations (see Marsden et al. [1999, 2000a]),

and the Discrete Routh equations (see Jalnapurkar et al. [2003]). The discrete analogues of exterior

3

calculus and curvature that arose in the development of discrete reduction (see Leok [2002]) moti-

vated us to systematically develop a discrete theory of exterior calculus (see Desbrun et al. [2003a]).

Similarly, attempts to develop a discrete analogue of the continuous theory of Lagrangian reduction

(see, for example, Cendra et al. [2001]) lead to the construction of discrete connections on principal

bundles (see Leok et al. [2003]). In addition, the construction of G-invariant discrete Lagrangians

suitable for discrete reduction motivated the work on generalized Galerkin variational integrators

(see Leok [2004]).

In the rest of this chapter, we will provide an overview of the material presented in this thesis.

Discrete Routh Reduction. We consider the problem of a discrete Lagrangian system with

an abelian symmetry group. By the discrete Noether’s theorem, the dynamics is restricted to a

constant discrete momentum surface J−1d (µ). By symmetry considerations, the dynamics can be

further restricted to J−1(µ)/G. We first construct a semi-global isomorphism between J−1d (µ)

and S × S using a discrete mechanical connection. This choice of discrete connection allows the

reconstruction of the reduced trajectory on S × S to the full trajectory on Q×Q to be interpreted

as a discrete horizontal lift.

By using the connection to split the variations in the discrete Hamilton’s principle into horizontal

and vertical variations, we drop the variational principle on Q×Q to the reduced variational principle

on S×S. The discrete equations corresponding to the reduced variational principle are the discrete

Routh equations, which are symplectic with respect to a reduced symplectic form that includes a

magnetic term arising from the curvature of the connection. This is particularly significant, since

standard symplectic methods can only preserve the canonical symplectic form, and discrete Routh

equations are the first numerical scheme that have the correct conservation properties in the presence

of a non-canonical symplectic form.

On the Hamiltonian side, the analogue of cotangent bundle reduction yields the Reduced Sym-

plectic Partitioned Runge–Kutta (RSPRK) algorithm. This algorithm is also symplectic with respect

to the non-canonical symplectic form on the reduced space, and includes corrections to Symplectic

Runge–Kutta that involve the curvature of the connection. Both the RSPRK algorithm and the

discrete Routh equations are consistent with the continuous theory of Lagrangian and Hamiltonian

reduction, and they form a commutative cube of equations that are related by maps between the

discrete and continuous theory, projections from the unreduced to the reduced spaces, and the fiber

derivatives that go between the Lagrangian and Hamiltonian formulation.

Forced or dissipative systems can be addressed in the reduced framework by reducing the discrete

Lagrange–d’Alembert principle, and the resulting methods exhibit superior tracking of the energy

decay, as compared to traditional methods. Constraints are handled through the use of Lagrange

4

multipliers, and allow computations on manifolds to be realized using constraint surfaces in a linear

space, thereby resulting in additional computational efficiency.

These reduced algorithms are particularly relevant when studying phenomena, such as relative

equilibria and relative periodic orbits, that are associated with dynamical structures on the reduced

space.

Discrete Exterior Calculus. A theory of discrete exterior calculus on simplicial complexes of

arbitrary finite dimension is constructed. In addition to dealing with discrete differential forms as

cochains, discrete vector fields and the operators acting on these objects are introduced. The various

interactions between forms and vector fields (such as Lie derivatives) which are important in many

applications, including fluid dynamics, can be addressed. Previous attempts at discrete exterior

calculus have addressed only differential forms. The notion of a circumcentric dual of a simplicial

complex is also introduced. The importance of dual complexes in this field has been well understood,

but previous researchers have used barycentric subdivision or barycentric duals. It is shown that the

use of circumcentric duals is crucial in arriving at a theory of discrete exterior calculus that admits

both vector fields and forms.

In this framework, one can systematically recover discrete vector differential operators like the

divergence, gradient, curl and the Laplace–Beltrami operator. These can be thought of as general-

izations of mimetic difference operators (see, for example, Shashkov [1996]; Hyman and Shashkov

[1997b,a]) to unstructured meshes, without the need for interpolation. Instead, these discrete differ-

ential operators are realized as combinatorial operations on the mesh. Methods based on interpola-

tion through the use of Whitney forms can be found in Bossavit [1998] and Hiptmair [1999].

The exactness property of the discrete variational complex is critical in many applications. Along

these lines, a discrete Poincare lemma is proved in the context of discrete exterior calculus. Here,

a homotopy operator that is valid for a large class of unstructured meshes is constructed, and the

lemma holds globally for specific examples of regular triangulations and tetrahedralizations of R2

and R3.

Discrete exterior calculus (DEC) and discrete variational mechanics have an interesting rela-

tionship. In particular, discretizing the equations describing harmonic maps and electromagnetism

using DEC yield the same numerical scheme as that obtained from the discrete variational principle

corresponding to an action integral that is discretized using DEC. This implies that directly dis-

cretizing the equations for harmonic maps and electromagnetism yield numerical schemes that have

the structure-preservation properties of variational schemes.

We also consider extensions to dynamic problems by using the groupoid formulation of discrete

diffeomorphisms and flows, and introduce the push-forward and pull-back of discrete vector fields

5

and forms. A method for remeshing cochains is also introduced, and this provides the restriction

and prolongation operators necessary to apply DEC operators in multigrid computations.

Discrete Connections on Principal Bundles. We were motivated by applications to discrete

Lagrangian reduction to introduce the notion of a discrete connection as a splitting of Q × Q into

horizontal and vertical subspaces. The sense in which these horizontal and vertical subspaces are

complementary, and combine to recover Q×Q requires introducing a composition of horizontal and

vertical elements that extends the pair groupoid composition on Q×Q.

This choice of discrete horizontal and vertical subspaces is equivalent to the choice of a splitting

of the discrete Atiyah sequence. Equivalent representations of a discrete connection, such as the

discrete connection 1-form, and the discrete horizontal lift, are also considered. Computational

issues such as the order of approximation to a continuous connection are addressed.

The composition on Q×Q to compose horizontal and vertical elements can be further extended

by using the discrete connection. This is significant, since it extends the pair groupoid composition

on Q×Q in a manner that is consistent with the principal bundle structure of π : Q→ S, and allows

(q0, q1) to be composed with (q0, q1) provided πq1 = πq0. In the case when q1 = q0, this extended

composition reduces to the standard pair groupoid composition.

Given a discrete G-invariant Lagrangian in discrete mechanics, one can introduce a discrete

momentum map. From this, a discrete mechanical connection is constructed by requiring that a

pair of points are in the horizontal distribution if their discrete momentum is zero. By using the

theory of equivalent representations of the discrete connection, the discrete mechanical connection

can be represented as a discrete connection 1-form.

The applications of discrete connections include discrete Lagrangian reduction, geometric control

theory, and a discrete notion of a Riemannian manifold, and the associated Levi-Civita connection

and its curvature.

Generalized Variational Integrators. We consider a generalization of variational integrators

by recognizing that the fundamental object that is being discretized is the action integral, and this

is achieved by discretizing sections of the configuration bundle over a base space through the choice

of a finite-dimensional function space, and the choice of a numerical quadrature in approximating

the action integral.

By expanding the interpolatory spaces to include those that are not parameterized by their end-

point values, we obtain a general class of variational integrators that impose continuity at endpoints

through the use of Lagrange multipliers. When the degrees of freedom chosen include the endpoint

values, the Lagrange multipliers can be eliminated to recover the standard discrete Euler–Lagrange

equations, and recovers the theory of high-order variational integrators as a special case.

6

Nonlinear approximation techniques, which allow space-time mesh points to vary, are the natural

method of incorporating spatio-temporal adaptivity in variational integrators and naturally general-

ize Symplectic-Energy-Momentum integrators. In multiscale problems, the function space is chosen

to include solutions of the cell problem, corresponding to solving for the fast dynamics while keeping

the slow variables fixed. The highly oscillatory integral is then evaluated using Filon–Lobatto tech-

niques that are a class of exponentially fitted numerical quadrature schemes with many desirable

properties. The combination of these two choices allows the multiscale variational integrator to

exactly solve for the fast dynamics, and thereby achieve convergence rates that are independent of

the ratio of the fast and slow timescales.

In expanding the function spaces that can be handled using the variational framework, Lie group

variational methods, and pseudospectral variational methods can be constructed as well, which are

of particular relevance to quantum mechanical simulations.

7

Chapter 2

Discrete Routh Reduction

In collaboration with Sameer M. Jalnapurkar, Jerrold E. Marsden, and Matthew West.

Abstract

This chapter investigates the relationship between Routh symmetry reduction and

time discretization for Lagrangian systems. Within the framework of discrete vari-

ational mechanics, a discrete Routh reduction theory is constructed for the case of

abelian group actions, and extended to systems with constraints and non-conservative

forcing or dissipation. Variational Runge–Kutta discretizations are considered in de-

tail, including the extent to which symmetry reduction and discretization commute.

In addition, we obtain the Reduced Symplectic Runge–Kutta algorithm, which can be

considered a discrete analogue of cotangent bundle reduction. We demonstrate these

techniques numerically for satellite dynamics about the Earth with a non-spherical J2

correction, and the double spherical pendulum. The J2 problem is interesting because

in the unreduced picture, geometric phases inherent in the model and those due to

numerical discretization can be hard to distinguish, but this issue does not appear in

the reduced algorithm, and one can directly observe interesting dynamical structures.

The main point of the double spherical pendulum is to provide an example with a

nontrivial magnetic term in which our method is still efficient, but is challenging to

implement using a standard method.

2.1 Introduction

Given a mechanical system with symmetry, we can restrict the flow on the phase space to a level set

of the conserved momentum. This restricted flow induces a “reduced” flow on the quotient of this

8

level set by the subgroup of the symmetry group that acts on it. Thus we obtain a reduced dynamical

system on a reduced phase space. The process of reduction has been enormously important for many

topics in mechanics such as stability and bifurcation of relative equilibria, integrable systems, etc.

The purpose of the present work is to contribute to the development of reduction theory for

discrete time mechanical systems, using the variational formulation of discrete mechanics described

in Marsden and West [2001]. We also explore the relationship between continuous time reduction and

discrete time reduction, and discuss reduction for symplectic Runge–Kutta integration algorithms

and its relationship to the theory of discrete reduction.

The discrete time mechanical systems used here are derived from a discrete variational principle

on the discrete phase space Q × Q. Properties such as conservation of symplectic structure and

conservation of momentum follow in a natural way from the discrete variational principle, and the

discrete evolution map can thus be regarded as a symplectic-momentum integrator for a continuous

system.

The theory of discrete variational mechanics in the form we shall use it has its roots in the

optimal control literature in the 1960’s; see, for example, Jordan and Polak [1964], and Hwang and

Fan [1967]. It was formulated in the context of mechanics by Maeda [1981], Veselov [1988, 1991]

and Moser and Veselov [1991]. It was further developed by Wendlandt and Marsden [1997], and

Marsden and Wendlandt [1997], including a constrained formulation, and by Marsden et al. [1998],

who extended these ideas to multisymplectic partial differential equations. For a general overview

and many more references we refer to Marsden and West [2001].

Although symplectic integrators have typically only been considered for conservative systems,

in Kane et al. [2000] it was shown how the discrete variational mechanics can be extended to

include forced and dissipative systems. This yields integrators for non-conservative systems which

can demonstrate exceptionally good long-time behavior, and which correctly simulate the decay or

growth in quantities such as energy and momentum. The discrete mechanics for non-conservative

systems is discussed in §2.8.2, and it is shown how the discrete reduction theory can also handle

forced and dissipative systems.

The formulation of discrete mechanics in this paper is best suited for constructing structure

preserving integrators for mechanical systems that are specified in terms of a regular Lagrangian.

Jalnapurkar and Marsden [2003], building on the work of Marsden and West [2001], show how to

obtain structure-preserving variational integrators for mechanical systems specified in terms of a

Hamiltonian. This method can be applied even if the Hamiltonian is degenerate.

A complementary approach to the Routh theory of reduction used in this paper is that of Lie–

Poisson and Euler–Poincare reduction, where the dynamics of an equivariant system on a Lie group

can be reduced to dynamics on the corresponding Lie algebra. A discrete variational formulation of

9

this was given in Marsden et al. [1999, 2000a], Bobenko et al. [1998], and Bobenko and Suris [1999].

Eventually one will need to merge that theory with the theory in the present paper.

We shall now briefly describe the contents of each section of this paper. In §2.2 we give a summary

of some well known results on reduction for continuous-time mechanical systems with symmetry.

Specifically, we discuss Routh reduction and its relationship with the theory of cotangent bundle

reduction. In §2.3 we develop the theory of discrete reduction, which includes the derivation of a

reduced variational principle, and proof of the symplecticity of the reduced flow. We also discuss

in this section the relationship between continuous- and discrete-time reduction. In §2.4 we discuss

a link between the theory of discrete mechanics and symplectic Runge–Kutta algorithms. In §2.5,

we describe how our symplectic Runge–Kutta algorithm for a mechanical system with symmetry

can be reduced to obtain a reduced symplectic Runge–Kutta algorithm. In §2.6 we put together

in a coherent way the results of the previous sections. We also discuss how the original reduction

procedure of Routh [1877, 1884] relates to our results. In §2.8 we extend the theory of discrete

reduction to systems with constraints and external forces, and lastly, in §2.9 we present a numerical

example of satellite dynamics about an oblate Earth.

2.2 Continuous Reduction

In this section we discuss reduction of continuous mechanical systems, in both the Lagrangian and

Hamiltonian settings. Our purpose here is to fix notation and recall some basic results. For a

more detailed exposition, see Marsden and Scheurle [1993a,b], Holm et al. [1998], Jalnapurkar and

Marsden [2000], Marsden et al. [2000b], and Cendra et al. [2001] for Lagrangian reduction, and for

Hamiltonian reduction, see, for example, Abraham and Marsden [1978] as well as Marsden [1992]

for cotangent bundle reduction.

Suppose we have a mechanical system with configuration manifold Q, and let L : TQ→ R be a

given Lagrangian. Let q = (q1, . . . , qn) be coordinates on Q. The Euler–Lagrange (EL) equations

on TQ are

∂L

∂q− d

dt

∂L

∂q= 0. (2.2.1)

These equations define a flow on TQ if L is a regular Lagrangian, which we assume to be the

case. Let XE denote the vector field on TQ that corresponds to the flow. We have a Legendre

transformation, FL : TQ→ T ∗Q, defined by

FL : (q, q) 7→(q,∂L

∂q

).

10

The Hamiltonian H on T ∗Q is obtained by pushing forward the energy function E on TQ, which is

defined by

E(q, q) = 〈FL(q, q), q〉 − L(q, q).

We have a canonical symplectic structure ΩQ on T ∗Q. From ΩQ and H, we obtain the Hamiltonian

vector field XH on T ∗Q. A basic fact is that XH is the push-forward of XE using FL. Since the

flow of XH preserves ΩQ, the flow of XE , i.e., the flow derived from the EL equations, preserves

ΩL := (FL)∗ΩQ.

Suppose an abelian group G acts freely and properly on Q so that Q is a principal fibre bundle

over shape space S := Q/G. Let πQ,S : Q → S be the natural projection. Given x ∈ S, we

can find an open set U ⊂ S, such that π−1Q,S(U) is diffeomorphic to G × U . Such a diffeomorphism

is called a local trivialization. Given a local trivialization, we can use local coordinates on G and

on S to obtain a set of local coordinates on Q. If g = (g1, . . . , gr) and x = (x1, . . . , xs) are local

coordinates on G and U ⊂ S, respectively, then q = (g, x) = (g1, . . . , gr, x1, . . . , xs) can be taken as

local coordinates on Q.

The action of G on Q on be lifted to give actions of G on TQ and T ∗Q. We also have a momentum

map J : T ∗Q → g∗, defined by the equation J(αq) · ξ = 〈αq, ξQ(q)〉, where αq ∈ T ∗qQ, ξ ∈ g, and

ξQ(q) is the infinitesimal generator corresponding to the action of G on Q evaluated at q. We can

pull-back J to TQ using the Legendre transform FL to obtain a Lagrangian momentum map

JL := FL∗J : TQ→ g∗.

If the Lagrangian L is invariant under the lifted action of G on TQ, the associated Hamiltonian

H will be invariant under the action of G on T ∗Q. In this situation, Noether’s theorem tells us that

the flows on TQ and on T ∗Q preserve the momentum maps JL and J , respectively.

Since locally, Q ≈ G × S, we also have the local representation TQ ≈ TG × TS. Thus, if (g, g)

are local coordinates on TG, and (x, x) are local coordinates on TS, (g, x, g, x) are local coordinates

on TQ. From the formula for the momentum map and freeness of the action, one sees that g is

determined from (g, x, x) and the value of the momentum. Thus, J−1L (µ) is locally diffeomorphic to

G × TS. If G is abelian (which is what we have assumed), it follows that G acts on J−1L (µ), and

that J−1L (µ)/G is locally diffeomorphic to TS. Let the natural projection J−1

L (µ) → TS be denoted

by πµ,L.

In a local trivialization, let q ∈ Q correspond to (g, x) ∈ G×S. Thus TqQ can be identified with

TgG× TxS.

Let A : TQ → g be a chosen principal connection. Using a local trivialization, the connection

11

can be described by the equation

A(g, x) = A(x)x+ g−1 · g.

Here A(x) : TxS → g is the restriction of A to TxS. (TxS is identified with the subspace TxS × 0

of TgG × TxS, which is in turn identified with TqQ.) Note that the map A(x) depends upon the

particular trivialization that we are using.

The connection gives us an intrinsic way of splitting each tangent space to Q into horizontal and

vertical subspaces. The vertical space Vq at q is the tangent space to the group orbit through q. If

Aq : TqQ → g is the restriction of A, then the horizontal space Hq is defined as the kernel of Aq.

The maps hor : TqQ→ Hq and ver : TqQ→ Vq are the horizontal and vertical projections obtained

from the split TqQ = Hq ⊕ Vq.

If L is of the form kinetic minus potential energy, then A can be chosen to be the mechanical

connection, although we shall not insist on this choice. However, in this case one gets, for example,

as in Marsden et al. [2000b], a global diffeomorphism J−1L (µ)/G ∼= TS.

Reduction on the Lagrangian Side. From the connection A we obtain a 1-form Aµ on Q defined

by

Aµ(q)q := 〈µ,A(q)〉.

The exterior derivative dAµ of Aµ is a 2-form on Q. It can be shown (see, for example, Marsden

[1992] or Marsden et al. [2000b]) that dAµ is G-invariant and is zero on all vertical tangent vectors

to Q. Thus, dAµ drops to a 2-form on S, which we shall call βµ. It is often called the magnetic

2-form.

If q is a curve that solves the Euler–Lagrange equations, then it is a solution of Hamilton’s

variational principle, which states that

δ

∫ b

a

L(q, q) dt = 0,

for all variations δq of q that vanish at the endpoints. The curve x obtained by projecting this

solution q onto the shape space also solves a variational principle on the shape space. This reduced

variational principle has the form

δ

∫ b

a

Rµ(x, x) dt =∫ b

a

βµ(x, δx) dt, (2.2.2)

for all variations δx of x that vanish at the endpoints and for a function Rµ that we shall now define.

12

To define the Routhian Rµ on TS, we first define a function Rµ on TQ by

Rµ(q, q) = L(q, q)− Aµ(q)q,

where µ is the momentum of the solution q. The restriction of Rµ to J−1L (µ) is G-invariant and Rµ

is obtained by dropping Rµ|J−1L (µ) to J−1

L (µ)/G ≈ TS.

It is easy to check that the reduced variational principle above is equivalent to the equations

∂Rµ

∂x− d

dt

∂Rµ

∂x= ixβµ(x), (2.2.3)

where ix denotes interior product of the 2-form βµ with the vector x. We call Equation 2.2.3 the

Routh equations.

Reduction on the Hamiltonian Side. If the group G is abelian (which is what we have as-

sumed), then from equivariance of the momentum map, we see thatG acts on the momentum level set

J−1(µ) ⊂ T ∗Q. The quotient J−1(µ)/G can be identified with T ∗S. The projection J−1(µ) → T ∗S

called πµ and can be defined as follows: If αq ∈ J−1(µ), then the momentum shift αq − Aµ(q)

annihilates all vertical tangent vectors at q ∈ Q, as shown by the following calculation:

〈αq − Aµ(q), ξQ(q)〉 = J(αq) · ξ − 〈µ, ξ〉 = 〈µ, ξ〉 − 〈µ, ξ〉 = 0.

Thus, αq − Aµ(q) induces an element of T ∗xS and πµ(αq) is defined to be this element.

By Noether’s theorem, the flow of the Hamiltonian vector fieldXH leaves the set J−1(µ) invariant

and is equivariant, and so the restricted flow induces a reduced flow on T ∗S. This reduced flow

corresponds to a reduced Hamiltonian vector XHµon T ∗S, which can be obtained from a reduced

Hamiltonian Hµ and a reduced symplectic form Ωµ. The reduced energy at momentum level µ is

denoted Hµ and is obtained by restricting H to J−1(µ) and then, using its invariance, to drop it to

a function on T ∗S. Similarly, we get the reduced symplectic form Ωµ by restricting ΩQ to J−1(µ)

and then dropping to T ∗S; namely, the reduced symplectic structure Ωµ is related to ΩQ by the

equation

π∗µΩµ = i∗µΩQ,

and is preserved by the reduced flow. An important result for cotangent bundles is that Ωµ =

ΩS − π∗T∗S,Sβµ, where ΩS is the canonical symplectic form on T ∗S, and πT∗S,S : T ∗S → S is the

natural projection. See, for example, Marsden [1992] for the proof.

13

Relating Lagrangian and Hamiltonian Reduction. The projections πµ,L : J−1L (µ) → TS and

πµ : J−1(µ) → T ∗S are related by the equation,

πµ FL = FRµ πµ,L,

where FRµ : TS → T ∗S is the reduced Routh-Legendre transform and is defined by

FRµ : (x, x) 7→

(x,∂Rµ

∂x

).

Notice that the Routhian Rµ has the momentum shift built into it as does the projection πµ. It

readily follows that the reduced dynamics on TS and on T ∗S, given by the Routh equations and the

vector field XHµ, respectively, are also related by the reduced Legendre transform FRµ. Thus, the

relationships between the reduced and “unreduced” spaces and the reduced and unreduced dynamics

can be depicted in the following commutative diagram:

(J−1L (µ), EL) FL //

πµ,L

(J−1(µ), XH)

πµ

(TS,R) FRµ// (T ∗S,XHµ)

From the commutativity of this diagram, one sees that conservation of the symplectic 2-form

(FRµ)∗(ΩS − π∗T∗S,Sβµ) by the flow of the Routh equations follows from the conservation of the 2-

form ΩS−π∗T∗S,Sβµ by the flow of the reduced Hamiltonian vector field. Conservation of (FRµ)∗(ΩS−

π∗T∗S,Sβµ) can also be shown directly from the reduced variational principle (Equation 2.2.2).

Reconstruction. There is a general theory of reconstruction for both the Hamiltonian and La-

grangian sides of reduction. The problem is this: given an integral curve in the reduced space TS

or T ∗S, a value of µ and an initial condition in the µ-level set of the momentum map, how does one

reconstruct the integral curve through that initial condition in TQ or T ∗Q? This question involves

the theory of geometric phases and of course is closely related to the classical constructions of so-

lutions by quadratures given a set of integrals of motion. This is not a trivial procedure, even for

abelian symmetry groups, although in this case things are somewhat more explicit. This procedure

is discussed at length in, for example, Marsden et al. [1990], Marsden [1992] and Marsden et al.

[2000b]. We shall need this theory at a couple of points in what follows.

14

2.3 Discrete Reduction

2.3.1 Discrete Variational Mechanics

In this paper, we will be using the theory of discrete mechanics as described in Marsden and West

[2001]. In this subsection, we briefly describe the essential features of this theory and fix our notation.

Discrete Lagrangians. Given a configuration manifold Q, a discrete Lagrangian system consists

of the discrete phase space Q × Q and a discrete Lagrangian Ld : Q × Q → R. As we are

interested in discrete systems which approximate a given continuous system, we will take discrete

Lagrangians which depend on a timestep h, so that Ld : Q×Q× R → R should be thought of as

approximating the action for time h,

Ld(q0, q1, h) ≈∫ h

0

L(q(t), q(t)) dt, (2.3.1)

where q : [0, h] → Q is a continuous trajectory solving the Euler–Lagrange equations for L with

boundary conditions q(0) = q0 and q(h) = q1. When the timestep is fixed in a discussion, we often

neglect the timestep dependence in Ld and write Ld(q0, q1) for simplicity.

Discrete Euler–Lagrange Equations. Just as continuous trajectories are maps from [0, T ] to

Q, we consider discrete trajectories, which are maps from 0, h, 2h, . . . , Nh = T to Q. This gives

a set of points in Q which we denote q = qkNk=0.

Having defined a discrete Lagrangian, we define the discrete action to be a function mapping

discrete trajectories q = qkNk=0 to the reals, given by

Gd(q) =N−1∑k=0

Ld(qk, qk+1). (2.3.2)

Hamilton’s principle requires that the discrete action be stationary with respect to variations van-

ishing at k = 0 and k = N . Computing the variations gives

dGd(q) · δq =N−1∑k=0

[D1Ld(qk, qk+1) · δqk +D2Ld(qk, qk+1) · δqk+1]

=N−1∑k=1

[D2Ld(qk−1, qk) +D1Ld(qk, qk+1)] · δqk

+D1Ld(q0, q1) · δq0 +D2Ld(qN−1, qN ) · δqN .

The requirement that this be zero for all variations satisfying δq0 = δqN = 0 gives the discrete

15

Euler–Lagrange (DEL) equations,

D2Ld(qk−1, qk) +D1Ld(qk, qk+1) = 0, (2.3.3)

for each k = 1, . . . , N−1. These implicitly define the discrete Lagrange map, FLd: Q×Q→ Q×Q;

(qk−1, qk) 7→ (qk, qk+1). We also refer to this map as the discrete Lagrangian evolution operator.

Discrete Lagrange Forms. The boundary terms in the expression for dGd can be identified as

the two discrete Lagrange 1-forms on Q×Q, which are

Θ+Ld

(q0, q1) = D2Ld(q0, q1)dq1, (2.3.4a)

Θ−Ld

(q0, q1) = −D1Ld(q0, q1)dq0. (2.3.4b)

In coordinates, note that

Θ+Ld

=∂Ld∂qi1

dqi1. (2.3.5)

We define the discrete Lagrange 2-form on Q×Q to be

ΩLd= −dΘ+

Ld, (2.3.6)

which in coordinates is

ΩLd= −d

(∂Ld∂qi1

dqi1

)=

∂2Ld

∂qi1∂qj2

dqi1 ∧ dqj2. (2.3.7)

A straightforward calculation shows that

ΩLd= −dΘ−

Ld. (2.3.8)

The space of solutions of the discrete Euler–Lagrange equations can be identified with the space

Q×Q of initial conditions (q0, q1). Restricting the free variations of the discrete action to this space

shows that we have

dGd|Q×Q = −Θ−Ld

+ (FN−1Ld

)∗(Θ+Ld

),

and so taking a second derivative and using the fact that d2 = 0 shows that

(FN−1Ld

)∗ΩLd= ΩLd

.

16

In particular, taking N = 2, we see that the discrete Lagrange evolution operator is symplectic; that

is,

(FLd)∗ΩLd

= ΩLd. (2.3.9)

Discrete Legendre Transforms. Given a discrete Lagrangian we define the discrete Legendre

transforms, F+Ld,F−Ld : Q×Q→ T ∗Q, by

F−Ld(q0, q1) = (q0,−D1Ld(q0, q1)), (2.3.10a)

F+Ld(q0, q1) = (q1, D2Ld(q0, q1)), (2.3.10b)

and we observe that the discrete Lagrange 1- and 2-forms are related to the canonical 1- and 2-

forms on T ∗Q by pull-back under the discrete Legendre transforms; that is, Θ±Ld

= (F±Ld)∗(Θ) and

ΩLd= (F±Ld)∗(Ω).

Pushing the discrete Lagrange map forward to T ∗Q with the discrete Legendre transform gives

the push-forward discrete Lagrange map, FLd: T ∗Q→ T ∗Q by FLd

= F±Ld FLd (F±Ld)−1.

One checks that one has the same map for the + case and the − case. In fact, the expression for

the push-forward discrete Lagrange map can be seen to be determined as follows: FLd: (q0, p0) 7→

(q1, p1), where

p0 = −D1Ld(q0, q1), (2.3.11a)

p1 = D2Ld(q0, q1). (2.3.11b)

Note that by construction, the push-forward discrete Lagrange map preserves the canonical 2-form.

The push-forward discrete Lagrange map is thus symplectic; that is, (FLd)∗(Ω) = Ω.

Exact Discrete Lagrangians. The relationship between a discrete Lagrangian and the corre-

sponding push-forward discrete Lagrange map is that of generating functions of the first kind.

Generating function theory shows that for any symplectic map T ∗Q→ T ∗Q (at least those near the

identity), there is a corresponding generating function Q×Q→ R which generates the map in the

sense of Equation 2.3.11.

It is thus clear that there is a discrete Lagrangian for every symplectic map, including the exact

flow F tH : T ∗Q → T ∗Q of the Hamiltonian system corresponding to the Lagrangian L. This is

referred to as the exact discrete Lagrangian and Hamilton–Jacobi theory shows that it is equal

to the action,

LEd (q0, q1, h) =∫ h

0

L(q(t), q(t)) dt, (2.3.12)

17

where q : [0, h] → Q solves the Euler–Lagrange equations for L with q(0) = q0 and q(h) = q1. This

classical theorem of Jacobi is proved in, for example, Marsden and Ratiu [1999].

Using this exact discrete Lagrangian, the push-forward discrete Lagrange map will be exactly

the Hamiltonian flow map for time h, so that FLEd

= FhH . That is, discrete trajectories q = qkNk=0

will exactly sample continuous trajectories q(t), namely qk = q(kh).

Approximate Discrete Lagrangians. If we choose a discrete Lagrangian which only approxi-

mates the action, then the resulting push-forward discrete Lagrange map will only approximate the

true flow. The orders of approximation are related, so that if the discrete Lagrangian is of order r,

Ld =∫ h

0

L(q, q) dt,+O(hr+1), (2.3.13)

then the push-forward discrete Lagrange map will also be of order r; that is,

FhLd= FhH +O(hr+1). (2.3.14)

By choosing discrete Lagrangians which are at least consistent (r ≥ 1) we can regard the discrete

Lagrange map as an integrator for the continuous system.

2.3.2 Discrete Mechanical Systems with Symmetry

Let G be an abelian Lie group that acts freely and properly on the configuration manifold Q. We

will assume that our discrete Lagrangian Ld is invariant under the diagonal action of G on Q×Q.

Such a discrete Lagrangian could have been obtained by discretizing a continuous Lagrangian L that

is invariant under the lifted action of G on TQ. For a discussion of how to construct G-invariant

discrete Lagrangians from G-invariant Lagrangians using natural charts, please see §5.3.2.

Note that Q is a bundle over the shape space S = Q/G. Using a local trivialization, Q can be

locally identified with G× S. Thus Q×Q ≈ G× S ×G× S. With this identification, (qk, qk+1) =

(hk, xk, hk+1, xk+1). We will use ∂/∂gi, i = 1, 2, to denote partial derivatives with respect to the

first and second group variables, and ∂/∂si, i = 1, 2, to denote partial derivatives with respect to

the first and second shape space variables.

Given a discrete Lagrangian, the discrete momentum map, Jd : Q×Q→ g∗, is defined by

Jd(q0, q1) · ξ = D2Ld(q0, q1) · ξQ(q1). (2.3.15)

18

Since Ld is invariant under the action of G, we have

D1Ld(q0, q1) · ξQ(q0) +D2Ld(q0, q1) · ξQ(q1) = 0.

Thus,

Jd(q0, q1) · ξ = D2Ld(q0, q1) · ξQ(q1) = −D1Ld(q0, q1) · ξQ(q0)

= ξQ(q1) Θ+Ld

(q0, q1) = ξQ(q0) Θ−Ld

(q0, q1),

where X ω denotes the interior product of a vector X with a 1-form ω. Thus, if q0, q1, q2, . . .

solves the DEL equations, then

Jd(q1, q2) · ξ = D2Ld(q1, q2) · ξQ(q2) = −D1Ld(q1, q2) · ξQ(q1)

= D2Ld(q0, q1) · ξQ(q1) = Jd(q0, q1) · ξ.

Thus, the discrete momentum is conserved by the discrete Lagrange map, FLd: Q × Q → Q × Q,

FLd: (q0, q1) 7→ (q1, q2). In other words, the discrete momentum is conserved along solutions of the

DEL equations, which is referred to as the discrete Noether theorem.

By definition of Jd,

Jd(q0, q1) · ξ = J(D2Ld(q0, q1)) · ξ,

where J : T ∗Q→ g∗ is the momentum map on T ∗Q. Thus,

Jd = J FLd,

where FLd = D2Ld : Q×Q→ T ∗Q is the discrete Legendre transform. (Note that in §2.3.1 we had

two discrete Legendre transforms, F+Ld and F−Ld. For the remainder of this paper, we use the term

discrete Legendre transform and the symbol FLd to denote F+Ld to make a specific choice.) Thus,

FLd maps J−1d (µ), which is the µ-level set of the discrete momentum to J−1(µ). Also, since Jd is

conserved by the discrete evolution operator FLd, it follows that the push-forward discrete Lagrange

map FLd: T ∗Q→ T ∗Q preserves J .

In a local trivialization, where q1 = (θ1, x1),

ξQ(q1) =d

dt

t=0

(exp (tξ) · θ1, x1) = (TRθ1 · ξ, 0),

19

where Rθ1 denotes right multiplication on G by θ1. Thus,

Jd(q0, q1) · ξ =[∂Ld∂g2

∂Ld∂s2

]·

TRθ1 · ξ

0

=∂Ld∂g2

TRθ1 · ξ.

Hence,

Jd(q0, q1) =∂Ld∂g2

(θ0, x0, θ1, x1) TRθ1 .

The momentum map, Jd : Q×Q→ g∗, is equivariant as the following calculation shows:

Jd(θ · q0, θ · q1) · ξ = D1Ld(θ · q0, θ · q1) · ξQ(θ · q0)

= D1Ld(θ · q0, θ · q1) · θ · (Adθ−1 ξ)Q(q0)

= D1Ld(q0, q1) · (Adθ−1 ξ)Q(q0)

= Jd(q0, q1) Adθ−1 ·ξ

= (Ad∗θ−1 Jd(q0, q1)) · ξ.

Thus, the coadjoint isotropy subgroup Gµ of G acts on J−1d (µ). Since G is abelian, Gµ = G, and

thus G acts on J−1d (µ).

If the value of the momentum is µ, the equation

∂Ld∂g2

(θ0, x0, θ1, x1) TRθ1 = µ,

determines θ1 implicitly as a function of θ0, x0, x1 and µ. Thus the level set J−1d (µ) can be (locally)

identified with G× S × S. The quotient J−1d (µ)/G is thus locally diffeomorphic to S × S.

If we choose a momentum µ, it follows from the above discussion that there is a unique map

ψµ : S × S → G, such that,

Jd(e, xk, ψµ(xk, xk+1), xk+1) = µ.

Further, if θk ∈ G, θk · (e, xk, ψµ(xk, xk+1), xk+1) = (θk, xk, θk · ψµ(xk, xk+1), xk+1) is also in

J−1d (µ). Thus for a given µ, the function giving θk+1 in terms of θk, xk and xk+1 is

θk+1 = θk · ψµ(xk, xk+1). (2.3.16)

Reconstruction. The following lemma gives a basic result on the reconstruction of discrete curves

in the configuration manifold Q from those in the shape space S. The lemma is similar to its

20

continuous counterpart, as in Lemma 2.2 of Jalnapurkar and Marsden [2000]. Recall that Vq denotes

the vertical space at q, which is the space of all vectors at q that are infinitesimal generators ξQ(q) ∈

TqQ. We say that the discrete Lagrangian Ld is group-regular if the bilinear map D2D1Ld(q, q) :

TqQ × TqQ → R restricted to the subspace Vq × Vq is nondegenerate. Besides regularity, we shall

make group-regularity a standing assumption in this chapter as well.

Lemma 2.1 (Reconstruction Lemma). Let µ ∈ g∗ be given, and x = x0, . . . , xn be a suffi-

ciently closely spaced discrete curve in S. Let q0 ∈ Q be such that πQ,S(q0) = x0. Then, there is a

unique closely spaced discrete curve q = q0, . . . , qn such that πQ,S(qk) = xk and Jd(qk, qk+1) = µ,

for k = 0, . . . , n− 1.

Proof. We must construct a point q1 close to q0 such that πQ,S(q1) = x1 and Jd(q0, q1) = µ. The

construction of the subsequent points q2, . . . , qn will follow in an inductive fashion.

To do this, pick a local trivialization of the bundle πQ,S : Q → Q/G, where Q ≈ G × S locally,

and write points in this trivialization as qk = (θk, xk).

Given the point q0 = (θ0, x0), we seek a near identity group element g, such that q1 := (gθ0, x1)

satisfies Jd(q0, q1) = µ. By the definition of the discrete momentum map (Equation 2.3.15), this

means that we must satisfy the condition

D2Ld(q0, q1) · ξQ(q1) = 〈µ, ξ〉

for all ξ ∈ g. In the local trivialization, this means that

D2Ld((θ0, x0), (gθ0, x1)) · (TRgθ0ξ, 0) = 〈µ, ξ〉 ,

where Rg denotes right translation on the group by the element g.

Consider solving the above equation for θ1 = gθ0 as a function of θ0, x0, x1, with µ fixed. We

know the base solution corresponding to the case x1 = x0, namely g = e. The implicit function

theorem tells us that when x1 is moved away from x0, there will be a unique solution for g near

the identity, provided that the derivative of the defining relation with respect to g at the identity is

invertible. But this condition is a consequence of group-regularity, so the result follows.

Note that the above lemma makes no hypotheses about the sequence satisfying the discrete

Euler–Lagrange equations.

To obtain the reconstruction equation in the continuous case, we require that the lifted curve

is second-order, on the momentum surface, and that it projects down to the reduced curve. It is

appropriate to consider the discrete analogue of the second-order curve condition, since it may not

be apparent where we imposed such a condition.

21

We consider a discrete curve x as a sequence of points, (x0, x1), (x1, x2), . . . , (xn−1, xn) in S×S.

Lift each of the points in S × S to the momentum surface J−1d (µ) ⊂ Q × Q. This yields the

sequence, (q00 , q01), (q10 , q

11), . . . , (qn−1

0 , qn−11 ), which is unique up to a diagonal group action on Q×Q.

The discrete analogue of the second-order curve condition is that this sequence in Q ×Q defines a

discrete curve in Q, which corresponds to requiring that qk1 = qk+10 , for k = 0, . . . , n − 1, which is

clearly possible in the context of the reconstruction lemma.

This discussion of the discrete reconstruction equation naturally leads to issues of geometric

phases, and it would be interesting to obtain an expression for the discrete geometric phase in terms

of the discrete curve on shape space.

Reconstruction of Tangent Vectors. Let (q0, q1) be a lift of (x0, x1) to J−1d (µ), and (δx0, δx1)

be a tangent vector to S × S at (x0, x1). Given δq0 ∈ Tq0Q, with TπQ,S · δq0 = δx0, it is possible

to find a δq1 ∈ Tq1Q, with TπQ,S · δq1 = δx1, such that (δq0, δq1) is a tangent vector to J−1d (µ)

at (q0, q1). Indeed, if in a local trivialization, δq0 = (δθ0, δx0), then the required δq1 is (δθ1, δx1),

where δθ1 is obtained by differentiating Equation 2.3.16 as follows:

δθ1 = δθ0 · ψµ(x0, x1) + θ0 ·D1ψµ(x0, x1)δx0 + θ0 ·D2ψµ(x0, x1)δx1.

Discrete Connection. It should be noted that although our discussion of reconstruction is cast in

terms of local trivializations, it is in fact intrinsic and can be thought of as a discrete horizontal lift

in the sense of discrete connections developed in Chapter 4. The discrete connection associated with

the reconstruction to the discrete µ-momentum surface is represented by the discrete connection

1-form Ad : Q × Q → G, defined on a G-invariant neighborhood of the diagonal by Ad(q0, q1) = e

iff Jd(q0, q1) = µ, and extended to other points by

Ad(g0q0, g1q1) = g1g−10 .

The reconstruction lemma (Lemma 2.1) may be viewed as providing the horizontal lift of this discrete

connection.

The discrete connection given above is the natural choice of connection on Q×Q for the purpose

of constructing a unified formulation of discrete, Lagrangian, and Hamiltonian reduction. Recall the

following diagram,

(TQ, JL) FL // (T ∗Q, J)

(Q×Q, Jd)

FLd

OO

22

and consider the horizontal space on TQ given by the µ-momentum surface, J−1L (µ). Since JL =

(FL)∗J , and Jd = (FLd)∗J , it follows that (FL)∗J−1L (µ) = (FLd)∗J−1

d (µ), and as a consequence,

(FLd)∗(FL)∗J−1L (µ) = J−1

d (µ). This implies that the discrete reconstruction equation is simply the

horizontal lift with respect to the discrete connection on Q×Q that is consistent with the connection

on TQ with respect to the fiber derivatives FL and FLd, and is therefore an intrinsic operation. The

discrete connection obtained in this way is related to the discrete mechanical connection, and is

given by the discrete connection 1-form introduced above.

Discrete connections also yield a semi-global isomorphism (Q×Q)/G ∼= (S×S)⊕ G (see §4.4.8)

for neighborhoods of the diagonal, and this induces a semi-global isomorphism J−1d (µ)/G ∼= S × S,

which is a discrete analogue of the global diffeomorphism, J−1L (µ)/G ∼= TS, that was obtained in

Marsden et al. [2000b] with the use of the mechanical connection.

2.3.3 Discrete Reduction

In this section, we start by assuming that we have been given a discrete Lagrangian, Ld : Q×Q→ R,

that is invariant under the action of an abelian Lie group G on Q×Q.

Let q := q0, . . . , qn be a solution of the discrete Euler–Lagrange (DEL) equations. Let the value

of the discrete momentum along this trajectory be µ. Let xi = πQ,S(qi), so that x := x0, . . . , xn

is a discrete trajectory on shape space. Since q satisfies the discrete variational principle, it is

appropriate to ask if there is a reduced variational principle satisfied by x.

An important issue in dropping the discrete variational principle to the shape space is whether

we require that the varied curves are constrained to lie on the level set of the momentum map. The

constrained approach is adopted in Jalnapurkar and Marsden [2000], and the unconstrained approach

is used in Marsden et al. [2000b]. In the rest of this section, we will adopt the unconstrained approach

of Marsden et al. [2000b], and will show that the variations in the discrete action sum evaluated

at a solution of the discrete Euler–Lagrange equation depends only on the quotient variations, and

therefore drops to the shape space without constraints on the variations.

By G-invariance of Ld, the restriction of Ld to J−1d (µ) drops to the quotient J−1

d (µ)/G ≈ S×S.

The function obtained on the quotient is called the reduced Lagrangian and is denoted Ld. Let

πµ,d : J−1d (µ) → S × S be the projection. Let (q0, q1) ∈ J−1

d (µ), and (δq0, δq1) ∈ T(q0,q1)J−1d (µ). If

πQ,S · qi = xi and TπQ,S · δqi = δxi, i = 0, 1, then πµ,d(q0, q1) = (x0, x1) and Tπµ,d · (δq0, δq1) =

(δx0, δx1). In this situation, we get Ld(q0, q1) = Ld(x0, x1), and so

DLd(q0, q1) · (δq0, δq1) = DLd(x0, x1) · (δx0, δx1). (2.3.17)

For q a solution of the DEL equations, and x the corresponding curve on the shape space

23

S, let δx = ddε

ε=0

xε be a variation of x. Let δq = ddε

ε=0

qε be any variation of q such that

TπQ,S · δqi = δxi. Then,

δn−1∑k=0

Ld(qk, qk+1) =d

dε

ε=0

∑k

Ld(qkε, qk+1ε)

=∑k

DLd(qk, qk+1) · (δqk, δqk+1)

= D1Ld(q0, q1) · δq0

+n−1∑k=1

(D2Ld(qk−1, qk) +D1Ld(qk, qk+1)) · δqk

+D2Ld(qn−1, qn) · δqn

= D1Ld(q0, q1) · δq0 +D2Ld(qn−1, qn) · δqn, (2.3.18)

where we have used the fact that q satisfies the discrete Euler–Lagrange equations.

Recall that the discrete momentum map is given by

Jd(qk, qk+1) · ξ = D2Ld(qk, qk+1) · ξQ(qk+1) = −D1Ld(qk, qk+1) · ξQ(qk).

Given any connection A on Q, we have a horizontal-vertical split of each tangent space to Q.

Thus,

D2Ld(qn−1, qn) · δqn = D2Ld(qn−1, qn) · hor δqn +D2Ld(qn−1, qn) · ver δqn.

Now, ver δqn = ξQ(qn), where ξ = A(δqn). Thus, A(δqn) = A(ver δqn) = A(ξQ(qn)). So,

D2Ld(qn−1, qn) · ver δqn = D2Ld(qn−1, qn) · ξQ(qn)

= Jd(qn−1, qn) · ξ = 〈µ, ξ〉 = 〈µ,A(ξQ(qn))〉

= 〈µ,A(δqn)〉 = Aµ(qn) · δqn. (2.3.19)

Thus,

D2Ld(qn−1, qn) · δqn = D2Ld(qn−1, qn) · hor δqn + Aµ(qn) · δqn. (2.3.20)

Similarly,

D1Ld(q0, q1) · δq0 = D1Ld(q0, q1) · hor δq0 − Aµ(q0) · δq0. (2.3.21)

24

Thus, from Equation 2.3.18,

d

dε

ε=0

∑k

Ld(qkε, qk+1ε) =D1Ld(q0, q1) · hor δq0 +D2Ld(qn−1, qn) · hor δqn

+ Aµ(qn) · δqn − Aµ(q0) · δq0. (2.3.22)

Define a 1-form A on Q×Q by

A(q0, q1)(δq0, δq1) = Aµ(q1) · δq1 − Aµ(q0) · δq0. (2.3.23)

If π1, π2 : Q×Q→ Q are projections onto the first and the second components, respectively. Then,

A = π∗2Aµ − π∗1Aµ.

Using G-invariance of Aµ, it follows that A is G-invariant. Also,

A(q0, q1)(ξQ(q0), ξQ(q1)) = Aµ(q1) · ξQ(q1)− Aµ(q0) · ξQ(q0) = 〈µ, ξ〉 − 〈µ, ξ〉 = 0.

Thus, A annihilates all vertical tangent vectors to Q × Q. It is easy to check that the 1-form

A|J−1d (µ), obtained by restricting A to J−1

d (µ) is also G-invariant and annihilates vertical tangent

vectors to J−1d (µ). Therefore, A|J−1

d (µ) drops to a 1-form A on J−1d (µ)/G ≈ S × S.

If πµ,d : J−1d (µ) → J−1

d (µ)/G is the projection, and iµ,d : J−1d (µ) → Q×Q is the inclusion, then

A and A are related by the equation

π∗µ,dA = i∗µ,dA.

We define the 1-forms A+ and A− on S×S and the maps A1, A2 : S×S → T ∗S by the relations

A+(x0, x1) · (δx0, δx1) = A2(x0, x1) · δx1 = A(x0, x1) · (0, δx1),

A−(x0, x1) · (δx0, δx1) = A1(x0, x1) · δx0 = A(x0, x1) · (δx0, 0).

Note that we have the relations A = A+ + A−, and

A(x0, x1) · (δx0, δx1) = A1(x0, x1) · δx0 + A2(x0, x1) · δx1.

From Equation 2.3.23, it follows that,

Aµ(qn) · δqn − Aµ(q0) · δq0 =n−1∑k=0

Aµ(qk+1) · δqk+1 − Aµ(qk) · δqk

25

=n−1∑k=0

A(qk, qk+1) · (δqk, δqk+1). (2.3.24)

Thus, Equation 2.3.22 can be rewritten as

d

dε

ε=0

n−1∑k=0

Ld(qkε, qk+1ε) =D1Ld(q0, q1) · hor δq0 +D2Ld(qn−1, qn) · hor δqn

+n−1∑k=0

A(qk, qk+1) · (δqk, δqk+1), (2.3.25)

or equivalently,

n−1∑k=0

(DLd −A)(qk, qk+1) · (δqk, δqk+1) = D1Ld(q0, q1) · hor δq0 +D2Ld(qn−1, qn) · hor δqn. (2.3.26)

The following lemma shows the sense in which the 1-form (DLd − A) on Q × Q drops to the

quotient J−1d (µ)/G ≈ S × S.

Lemma 2.2. If (q0, q1) ∈ J−1d (µ) and (δq0, δq1) ∈ T(q0,q1)Q×Q with πQ,S(qi) = xi and TπQ,S ·δqi =

δxi, i = 0, 1, then

(DLd −A)(q0, q1) · (δq0, δq1) = (DLd − A)(x0, x1) · (δx0, δx1).

Proof. As we showed in the discussion at the end of §2.3.2, we can find δq′1 ∈ Tq1Q such that

TπQ,S · δq′1 = δx1 and (δq0, δq′1) ∈ T(q0,q1)J−1d (µ). Let δq1 = δq′1 + δq′′1 . Thus δq′′1 ∈ Tq1Q is vertical,

i.e., TπQ,S · δq′′1 = 0. Now,

(DLd −A)(q0, q1) · (δq0, δq1) = (DLd −A)(q0, q1) · (δq0, δq′1) + (DLd −A)(q0, q1) · (0, δq′′1 ).

Using Equation 2.3.17, and the fact that A|J−1d (µ) drops to a 1-form A on S × S, we get

(DLd −A)(q0, q1) · (δq0, δq′1) = (DLd − A)(x0, x1) · (δx0, δx1).

Also, by a calculation similar to that used to derive Equation 2.3.19, we have that

(DLd −A)(q0, q1) · (0, δq′′1 ) = D2Ld(q0, q1) · δq′′1 − Aµ(q1)δq′′1 = 0.

26

With this lemma, and Equation 2.3.26, we conclude that

n−1∑k=0

(DLd − A)(xk, xk+1) · (δxk, δxk+1) = D1Ld(q0, q1) · hor δq0 +D2Ld(qn−1, qn) · hor δqn.

(2.3.27)

If δx is a variation of x that vanishes at the endpoints, then hor δq0 = 0, and hor δq1 = 0. Therefore,

n−1∑k=0

(DLd − A)(xk, xk+1) · (δxk, δxk+1) = 0.

Equivalently,

δn−1∑k=0

Ld(xk, xk+1) =n−1∑k=0

A(xk, xk+1) · (δxk, δxk+1). (2.3.28)

Equating terms involving δxk on the left-hand side of Equation 2.3.28 to the corresponding terms

on the right, we get the discrete Routh (DR) equations giving dynamics on S × S:

D2Ld(xk−1, xk) +D1Ld(xk, xk+1) = A2(xk−1, xk) + A1(xk, xk+1). (2.3.29)

Note that these equations depend on the value of momentum µ.

Thus, we have shown that if q is a discrete curve satisfying the discrete Euler–Lagrange equations,

the curve x, obtained by projecting q down to S, satisfies the DR equations (Equation 2.3.29).

Now we shall consider the converse, the discrete reduction procedure: Given a discrete curve x

on S that satisfies the DR equations, is x the projection of a discrete curve q on Q that satisfies the

DEL equations?

Let the pair (q0, q1) be a lift of (x0, x1) such that Jd(q0, q1) = µ. Let q = q0, . . . , qn be the

solution of the DEL equations with initial condition (q0, q1). Note that q has momentum µ. Let

x′ = x′0, . . . , x′n be the curve on S obtained by projecting q. By our arguments above, x′ solves

the DR equations. However x′ has the initial condition (x0, x1), which is the same as the initial

condition of x. Thus, by uniqueness of the solutions of the DR equations, x′ = x. Thus x is the

projection of a solution q of the DEL equations with momentum µ. Also, for a given initial condition

q0, there is a unique lift of x to a curve with momentum µ. Such a lift can be constructed using the

method described in §2.3.2. Thus, lifting x to a curve with momentum µ yields a solution of the

discrete Euler–Lagrange equations, which projects down to x.

We summarize the results of this section in the following Theorem.

Theorem 2.3. Let x is a discrete curve on S, and let q be a discrete curve on Q with momentum

27

µ that is obtained by lifting x. Then the following are equivalent.

1. q solves the DEL equations.

2. q is a solution of the discrete Hamilton’s variational principle,

δn−1∑k=0

Ld(qk, qk+1) = 0,

for all variations δq of q that vanish at the endpoints.

3. x solves the DR equations,

D2Ld(xk−1, xk) +D1Ld(xk, xk+1) = A2(xk−1, xk) + A1(xk, xk+1).

4. x is a solution of the reduced variational principle,

δ∑k

Ld(xk, xk+1) =n−1∑k=0

A(xk, xk+1) · (δxk, δxk+1),

for all variations δx of x that vanish at the endpoints.

2.3.4 Preservation of the Reduced Discrete Symplectic Form

The DR equations define a discrete flow map, Fk : S × S → S × S. We already know that the flow

of the DEL equations preserves the symplectic form ΩLdon Q×Q. In this section we show that the

reduced flow Fk preserves a reduced symplectic form Ωµ,d on S×S, and that this reduced symplectic

form is obtained by restricting ΩLdto J−1

d (µ) and then dropping to S × S. In other words,

π∗µ,dΩµ,d = i∗µ,dΩLd.

The continuous analogue of this equation is

π∗µΩµ = i∗µΩQ.

Since the projections πµ,d and πµ involve a momentum shift, the reduced symplectic forms Ωµ,d and

Ωµ include magnetic terms.

Recall that Ld : S × S → R is the reduced Lagrangian, and DLd is a 1-form on S × S. Define

1-forms DL+d and DL−d on S × S by

DL+d (x0, x1) · (δx0, δx1) = DLd(x0, x1) · (0, δx1) = D2Ld(x0, x1) · δx1,

28

DL−d (x0, x1) · (δx0, δx1) = DLd(x0, x1) · (δx0, 0) = D1Ld(x0, x1) · δx0.

Note that the partial derivatives D1Ld and D2Ld are both maps S × S → T ∗S.

Define 2-forms B and B on Q×Q and S × S as follows:

B = dA, B = dA.

Since π∗µ,dA = i∗µ,dA, we get π∗µ,dB = i∗µ,dB. Thus B can be obtained by restricting B to J−1d (µ) and

then dropping to J−1d (µ)/G ≈ S × S.

Since A = π∗2Aµ − π∗1Aµ, it follows that B = π∗2Bµ − π∗1Bµ, where Bµ = dAµ is a 2-form on Q.

Now Bµ drops to a 2-form βµ on S. Using this fact, we find that B = π∗2βµ − π∗1βµ. Here,

π1, π2 : S × S → S are projections onto the first and second components, respectively. If we define

B+ := π∗2βµ, and B− := −π∗1βµ, then B = B− + B+.

We define a function S : S × S → R by

S(x0, x1) :=n−1∑k=0

Ld(xk, xk+1),

where x = x0, . . . , xn is a solution of the DR equations with initial condition (x0, x1). Thus,

S(x0, x1) =n−1∑k=0

Ld(Fk(x0, x1)).

Our goal in this section will be to show that symplecticity of the reduced flow follows from the fact

that d2S = 0.

Recall that if we lift x to a discrete curve q on Q with momentum µ, then q is a solution of the

DEL equations. Let (δx0, δx1) = ddε

ε=0

(x0ε, x1ε), and let xε = x0ε, . . . , xnε be a solution of the

discrete Routh equations with initial condition (x0ε, x1ε). Let δq be any variation of q such that

TπQ,S · δqi = δxi. Using Equation 2.3.27 in §2.3.3, we get

dS(x0, x1)(δx0, δx1) =d

dε

ε=0

S(x0ε, x1ε)

=d

dε

ε=0

n−1∑k=0

Ld(xkε, xk+1ε)

=n−1∑k=0

DLd(xk, xk+1) · (δxk, δxk+1)

= D1Ld(q0, q1) · hor δq0 +D2Ld(qn−1, qn) · hor δqn

29

+n−1∑k=0

A(xk, xk+1) · (δxk, δxk+1).

Thus,

dS(x0, x1)(δx0, δx1) =D1Ld(q0, q1) · hor δq0 +D2Ld(qn−1, qn) · hor δqn

+n−1∑k=0

(F ∗k A)(x0, x1) · (δx0, δx1). (2.3.30)

We will eventually prove conservation of a reduced symplectic form by taking the exterior derivative

of Equation 2.3.30. To do this, we need a number of preliminary calculations.

Lemma 2.4. d∑n−1

k=0(F ∗k A)

= (F ∗n−1B+ − B+)− B.

Proof. This is a straightforward verification using the facts that dA = B, and that

(F ∗k B)(x0, x1)((δx0, δx1), (δx′0, δx′1)) = B(xk, xk+1)((δxk, δxk+1), (δx′k, δx

′k+1))

= βµ(xk+1)(δxk+1, δx′k+1)− βµ(xk)(δxk, δx′k).

Here δxk, δx′k are obtained by pushing forward δx0, δx′0, respectively, and δxk+1, δx

′k+1 are obtained

by pushing forward δx1, δx′1, respectively.

Lemma 2.5.

D1Ld(q0, q1) · hor δq0 = −D2Ld(q0, q1) · hor δq1 + (DLd −A)(q0, q1) · (δq0, δq1).

Proof.

D1Ld(q0, q1) · hor δq0 =D1Ld(q0, q1) · δq0 −D1Ld(q0, q1) · ver δq0

=DLd(q0, q1) · (δq0, δq1)−D2Ld(q0, q1) · δq1 −D1Ld(q0, q1) · ver δq0

=DLd(q0, q1) · (δq0, δq1)−D2Ld(q0, q1) · hor δq1

−D2Ld(q0, q1) · ver δq1 −D1Ld(q0, q1) · ver δq0.

As in Equation 2.3.19,

D2Ld(q0, q1) · ver δq1 = Aµ(q1) · δq1.

30

Similarly,

D1Ld(q0, q1) · ver δq0 = −Aµ(q0) · δq0.

Thus,

D2Ld(q0, q1) · ver δq1 +D1Ld(q0, q1) · ver δq0 = A(q0, q1) · (δq0, δq1).

The statement of the lemma now follows.

Thus, Equation 2.3.30 can be rewritten as

dS(x0, x1)(δx0, δx1) =(D2Ld(qn−1, qn) · hor δqn −D2Ld(q0, q1) · hor δq1)

+n−1∑k=0

(F ∗k A)(x0, x1) · (δx0, δx1) + (DLd −A)(q0, q1) · (δq0, δq1)

=(D2Ld(qn−1, qn) · hor δqn −D2Ld(q0, q1) · hor δq1)

+n−1∑k=0

(F ∗k A)(x0, x1) · (δx0, δx1) + (DLd − A)(x0, x1) · (δx0, δx1). (2.3.31)

Lemma 2.6. D2Ld(q0, q1) · hor δq1 = ((DLd)+ − A+)(x0, x1) · (δx0, δx1).

Proof. Using Lemma 2.2, we get

D2Ld(q0, q1) · hor δq1 = D2Ld(q0, q1) · δq1 −D2Ld(q0, q1) · ver δq1

= DLd(q0, q1) · (0, δq1)− Aµ(q1) · δq1

= (DLd −A)(q0, q1) · (0, δq1)

= (DLd − A)(x0, x1) · (0, δx1)

= ((DLd)+ − A+)(x0, x1) · (δx0, δx1).

A consequence of this lemma is that

D2Ld(qn−1, qn) · hor δqn = ((DLd)+ − A+)(xn−1, xn) · (δxn−1, δxn)

= (F ∗n−1((DLd)+ − A+))(x0, x1) · (δx0, δx1).

Define the map F : S × S → T ∗S by

F(x0, x1) = D2Ld(x0, x1)− A2(x0, x1).

31

The map F will play the role of a discrete Legendre transform. Let ΘS be the canonical 1-form on

T ∗S.

Lemma 2.7. (DLd)+ − A+ = F∗ΘS .

Proof.

(F∗ΘS)(x0, x1) · (δx0, δx1) = ΘS(D2Ld(x0, x1)− A2(x0, x1)) · T F · (δx0, δx1)

= (D2Ld(x0, x1)− A2(x0, x1)) · TπS · T F · (δx0, δx1),

where πT∗S,S : T ∗S → S is the projection. Note that πT∗S,S F = π2, where π2 : S × S → S is the

projection onto the second component. Thus,

(F∗ΘS)(x0, x1) · (δx0, δx1) = (D2Ld(x0, x1)− A2(x0, x1)) · T π2 · (δx0, δx1)

= (D2Ld(x0, x1)− A2(x0, x1)) · δx1

= ((DLd)+ − A+)(x0, x1) · (δx0, δx1).

Using Lemmas 2.6 and 2.7, Equation 2.3.31 can be rewritten as:

dS = (F ∗n−1(F∗ΘS)− F∗ΘS) +n−1∑k=0

(F ∗k A) +DLd − A.

Taking the exterior derivative on both sides of this equation and using Lemma 2.4 and the fact that

d2 = 0 yields

0 = F ∗n−1(F∗ΩS − B+)− (F∗ΩS − B+),

where ΩS = −dΘS is the canonical 2-form on T ∗S. Since πT∗S,S F = π2,

B+ = π∗2βµ = F∗π∗T∗S,Sβµ.

Thus,

F∗ΩS − B+ = F∗(ΩS − π∗T∗S,Sβµ).

We have thus proved the following Theorem.

Theorem 2.8. The flow of the DR equations preserves the symplectic form

Ωµ,d = F∗ΩS − B+

= (D2Ld − A2)∗ΩS − π∗2βµ

32

= F∗(ΩS − π∗T∗S,Sβµ).

We remark that the symplectic form Ωµ,d is just the pull-back by F of the same symplectic form

on T ∗S that is obtained by the process of cotangent bundle reduction (see §2.2). The fact that Ωµ,d

is closed follows from the closure of (ΩS − π∗T∗S,Sβµ).

We will now complete our argument by showing that


We showed in section §2.3.2 that the discrete Legendre transform FLd : Q × Q → T ∗Q, (q0, q1) 7→

D2Ld(q0, q1) maps J−1d (µ) to J−1(µ), where Jd and J are the discrete and continuous momentum

maps, respectively. For the rest of this section, let F′ : J−1d (µ) → J−1(µ) be the restriction of FLd.

Thus F′ iµ = iµ,d FLd, where iµ : J−1(µ) → T ∗Q and iµ,d : J−1d (µ) → Q×Q are inclusions.

Recall that we had defined the map F : S × S → T ∗S as D2Ld − A2.

Lemma 2.9. The following diagram commutes.

J−1d (µ) F′ //

πµ,d

J−1(µ)

πµ

S × SF // T ∗S

Proof. Let (q0, q1) ∈ J−1d (µ). Thus D2Ld(q0, q1) ∈ J−1(µ), and

πµ(F′(q0, q1)) = πµ(D2Ld(q0, q1)).

Recall from §2.2 that (D2Ld(q0, q1) − Aµ(q1)) annihilates all vertical tangent vectors and that

πµ(D2Ld(q0, q1)) is the element of T ∗x1S determined by (D2Ld(q0, q1)− Aµ(q1)). For δq1 ∈ Tq1Q,

〈D2Ld(q0, q1)− Aµ(q1), δq1〉 = 〈D2Ld(q0, q1)− Aµ(q1),hor δq1〉.

Using the fact that Aµ(q1) annihilates horizontal vectors, and Lemma 2.6, we obtain

〈D2Ld(q0, q1)− Aµ(q1), δq1〉 = 〈D2Ld(q0, q1),hor δq1〉

= D2Ld(x0, x1) · δx1 − A2(x0, x1) · δx1 .

Thus,

πµ(D2Ld(q0, q1)) = D2Ld(x0, x1)− A2(x0, x1),

33

which means F πµ,d = πµ F′.

Using this lemma, we get

π∗µ,dΩµ,d = π∗µ,dF∗(ΩS − π∗T∗S,Sβµ)

= (F′)∗π∗µ(ΩS − π∗T∗S,Sβµ)

= (F′)∗i∗µΩQ = i∗µ,dFL∗dΩQ

= i∗µ,dΩLd.

Here, we have used the fact that π∗µ(ΩS − π∗βµ) = i∗µΩQ, which comes from the theory of cotangent

bundle reduction. We have thus proved the following Theorem.

Theorem 2.10. The flow of the DR equations preserves the symplectic form

Ωµ,d = F∗(ΩS − π∗T∗S,Sβµ).

Ωµ,d can be obtained by dropping to S × S the restriction of ΩLdto J−1

d (µ). In other words,


In proving Theorem 2.10, we started from the reduced variational equation (Equation 2.3.30).

There is also an alternate route to proving symplecticity of the reduced flow which relies on the fact

that discrete flow on Q×Q preserves the symplectic form ΩLd. We will give an outline of the steps

involved, without giving all the details. The idea is to first show that the restriction to J−1d (µ) of

the symplectic form ΩLddrops to a 2-form Ωµ,d on S ×S. The fact that the discrete flow on Q×Q

preserves the symplectic form ΩLdis then used to show that the reduced flow preserves Ωµ,d.

The outline of the steps involved is as follows.

1. Consider the 1-form ΘLdon Q×Q defined by ΘLd

(q0, q1) · (δq0, δq1) = D2Ld(q0, q1) · δq1. ΘLd

is G-invariant, and thus the Lie derivative LξQ×QΘLd

is zero.

2. Since ΩLd= −dΘLd

, ΩLdis G-invariant. If iµ,d : J−1

d (µ) → Q × Q is the inclusion, Θ′Ld

=

i∗µ,dΘLdand Ω′Ld

= i∗µ,dΩLdare the restrictions of ΘLd

and ΩLd, respectively, to J−1

d (µ). It is

easy to check that Θ′Ld

and Ω′Ldare invariant under the action of G on J−1

d (µ).

3. If ξJ−1d (µ) is an infinitesimal generator on J−1

d (µ), then

ξJ−1d (µ) Ω′Ld

= −ξJ−1d (µ) dΘ′

Ld= −Lξ

J−1d

(µ)Θ′Ld

+ dξJ−1d (µ) Θ′

Ld= 0.

34

This follows from the G-invariance of Θ′Ld

, and the fact that Θ′Ld· ξJ−1

d (µ) = 〈µ, ξ〉.

4. By steps 2 and 3, the form Ω′Lddrops to a reduced form Ωµ,d on J−1

d (µ)/G ≈ S × S. Thus, if

πµ,d : J−1d (µ) → S × S is the projection, then π∗µ,dΩµ,d = Ω′Ld

. Note that the closure of Ωµ,d

follows from the fact that Ω′Ldis closed, which in turn follows from the closure of ΩLd

and the

relation Ω′Ld= i∗µ,dΩLd

.

5. If Fk : Q×Q→ Q×Q is the flow of the DEL equations, let F ′k be the restriction of this flow

to J−1d (µ). We know that F ′k drops to the flow Fk of the DR equations on S × S. Since Fk

preserves ΩLd, F ′k preserves Ω′Ld

. Using this, it can be shown that Fk preserves Ωµ,d. Note

that it is sufficient to show that π∗µ,d(F∗kΩµ,d) = π∗µ,dΩµ,d.

6. It now remains to compute a formula for the reduced form Ωµ,d. Using Lemma 2.9 (whose

proof, in turn, relies on Lemma 2.6), it follows that

π∗µ,dΩµ,d = i∗µ,dΩLd= i∗µ,dFL∗dΩQ = (F′)∗i∗µΩQ

= (F′)∗π∗µ(ΩS − π∗T∗S,Sβµ)

= π∗µ,dF∗(ΩS − π∗T∗S,Sβµ).

Thus π∗µ,dΩµ,d = π∗µ,dF∗(ΩS−π∗T∗S,Sβµ), from which it follows that Ωµ,d = F∗(ΩS−π∗T∗S,Sβµ).

Incidentally, this expression shows that Ωµ,d is nondegenerate provided the map F = D2Ld−A2

is a local diffeomorphism.

2.3.5 Relating Discrete and Continuous Reduction

As we stated in §2.3.1, if the discrete Lagrangian Ld approximates the Jacobi solution of the

Hamilton–Jacobi equation, then the DEL equations give us an integration scheme for the EL equa-

tions. In our commutative diagrams we will denote the relationship between the EL and DEL

equations by a dashed arrow as follows:

(TQ,EL) //___ (Q×Q,DEL).

Thus, −− → can be read as “the corresponding discretization”. By the continuous and discrete

Noether theorems, we can restrict the flow of the EL and DEL equations to J−1L (µ) and J−1

d (µ),

respectively. We have seen that the flow on J−1L (µ) induces a reduced flow on J−1

L (µ)/G ≈ TS,

which is the flow of the Routh equations. Similarly, the discrete flow on J−1d (µ) induces a reduced

discrete flow on J−1d (µ)/G ≈ S×S, which is the flow of the discrete Routh equations. Since the DEL

equations give us an integration algorithm for the EL equations, it follows that the DR equations

35

give us an integration algorithm for the Routh equations.

Thus, to numerically integrate the Routh equations, we can follow either of the following ap-

proaches:

1. First solve the DEL equations to yield a discrete trajectory on Q, which can then be projected

to a discrete trajectory on S.

2. Solve the DR equations to directly obtain a discrete trajectory on Q.

Either approach will yield the same result. We can express this situation by the following commu-

tative diagram:

(J−1L (µ), EL) //___

πµ,L

(J−1d (µ), DEL)

πµ,d

(TS,R) //_____ (S × S,DR)

(2.3.32)

The upper dashed arrow represents the fact that the DEL equations are an integration algorithm

for the EL equations, and the lower dashed arrow represents the same relationship between the DR

equations and the Routh equations. Note that for smooth group actions the order of accuracy will be

equal for the reduced and unreduced algorithms. We will state this result precisely in the following

corollary.

Corollary 2.11. Given a discrete Lagrangian Ld : Q × Q → R of order r, and a smooth group

action, the discrete Routh equations associated with the reduced discrete Lagrangian, Ld : S×S → R,

obtained by dropping Ld to S × S, is of order r as well.

Proof. Recall that the order of the discrete Lagrangian is equal to the order of the push-forward

discrete Lagrangian map, and as such, the discrete Euler–Lagrange equations yield a r-th order

accurate approximation of the exact flow. When the group action is smooth, the projections πµ,L

and πµ,d are smooth as well. Since the two projections agree when restricted to the position space,

and the projections are smooth, the commutative diagram in Equation 2.3.32, together with the

chain rule, implies that the discrete Routh equations yield a r-th order accurate approximation to

the reduced flow.

2.4 Relating the DEL Equations to Symplectic Runge–Kutta

Algorithms

Symplectic Partitioned Runge–Kutta Methods. A well-studied class of numerical schemes

for Hamiltonian and Lagrangian systems is the partitioned Runge–Kutta (PRK) algorithms (see

36

Hairer et al. [1993] and Hairer and Wanner [1996] for history and details). Stated for a regular

Lagrangian system, a partitioned Runge–Kutta scheme is a map F : T ∗Q → T ∗Q defined by

F : (q0, p0) 7→ (q1, p1), where

q1 = q0 + hs∑j=1

bjQj , p1 = p0 + hs∑j=1

bjPj , (2.4.1a)

Qi = q0 + hs∑j=1

aijQj , Pi = p0 + hs∑j=1

aijPj , i = 1, . . . , s, (2.4.1b)

Pi =∂L

∂q(Qi, Qi), Pi =

∂L

∂q(Qi, Qi), i = 1, . . . , s, (2.4.1c)

where bi, bi, aij and aij are real coefficients for i, j = 1, . . . , s which define the method. Note that

Equation 2.4.1c implicitly determined the Hamiltonian vector field (Qi, Pi) at the point (Qi, Pi) =

FL(Qi, Qi).

The partitioned Runge–Kutta method, F : T ∗Q → T ∗Q, approximates the flow map, F tH :

T ∗Q→ T ∗Q, of the Hamiltonian system corresponding to the Lagrangian L, so that

F (q, p, h) = FhH(q, p) +O(hr+1),

where r, the order of the integration algorithm, is determined by the choice of the coefficients bi, bi,

aij and aij .

As discussed in §2.2, the flow map F tH of the Hamiltonian system on T ∗Q preserves the canonical

symplectic form Ω on T ∗Q. It can be shown that the partitioned Runge–Kutta method F preserves

the canonical symplectic form if, and only if, the coefficients satisfy

biaij + bjaji = bibj , i, j = 1, . . . , s (2.4.2a)

bi = bi, i = 1, . . . , s. (2.4.2b)

Such schemes are known as symplectic partitioned Runge–Kutta (SPRK) methods.

Discrete Lagrangians for SPRK Methods. For any given time-step h, a symplectic partitioned

Runge–Kutta method is a symplectic map F : T ∗Q→ T ∗Q. Therefore, as discussed in §2.3.1, there

is a discrete Lagrangian Ld which generates it.

An explicit form for this discrete Lagrangian was found by Suris [1990], and is given by

Ld(q0, q1, h) = hs∑i=1

biL(Qi, Qi),

37

where Qi, Qi, Pi and Pi are such that Equations 2.4.1b and 2.4.1c are satisfied. It can then be

shown, under assumptions (Equations 2.4.2a and 2.4.2b) on the coefficients, that the push-forward

of the discrete Lagrangian map is exactly the symplectic partitioned Runge–Kutta method. The

details of this calculation can be found in Suris [1990] or Marsden and West [2001].

For a partitioned Runge–Kutta method to be consistent, the coefficients must satisfy∑si=1 bi = 1.

With this in mind, it can be readily seen that the Ld defined above is an approximation to the action

over the interval [0, h], as one would expect from §2.3.1.

Discrete Lagrangians from Polynomials and Quadrature. While the discrete Lagrangian

given above generates any symplectic partitioned Runge–Kutta method, there is a subset of such

methods for which the discrete Lagrangian has a particularly elegant form. These can be derived

by approximating the action with polynomial trajectories and numerical quadrature.

As shown in §2.3.1, a discrete Lagrangian should be an approximation

Ld(q0, q1, h) ≈ extq∈C(0,h)

S(q),

where C(0, h) is the space of trajectories q : [0, h] → Q with q(0) = q0 and q(h) = q1, and S :

C(0, h) → R is the action S(q) =∫ h0L(q, q)dt.

To approximate this, we take a finite-dimensional approximation Cd(0, h) ⊂ C(0, h) of the trajec-

tory space,

Cd(0, h) = q ∈ C(0, h) | q is a polynomial of degree s,

and we approximate the action integral by numerical quadrature to give an approximate action

Sd : C(0, h) → R,

Sd(q) = hs∑i=1

biL(q(cih), q(cih)),

where (bi, ci) is the maximal-order quadrature rule on the unit interval with quadrature points ci.

We now set the discrete Lagrangian to be

Ld(q0, q1, h) = extqd∈Cd(0,h)

Sd(qd),

which can be explicitly evaluated. This procedure corresponds to the Galerkin projection of the

weak form of the ODE onto the space of piecewise polynomial trajectories, an interpretation which

is further discussed in Marsden and West [2001].

Theorem 2.12. Take a set of quadrature points ci and let Ld be the corresponding discrete La-

grangian as described above. Then the integrator generated by this discrete Lagrangian is equivalent

38

to the partitioned Runge–Kutta scheme defined by the coefficients

bi = bi =∫ 1

0

li,s(τ)dτ,

aij =∫ ci

0

lj,s(τ)dτ,

aij = bj

(1− aji

bi

),

(2.4.3)

where the li,s(τ) are the Lagrange polynomials associated with the ci.

Proof. Evaluating the conditions which imply that qd extremizes Sd and combining this with the

definition of the push-forward of the discrete Euler–Lagrange equations give the desired result. See

Marsden and West [2001] for details.

2.5 Reduction of the Symplectic Runge–Kutta Algorithm

Consider the SPRK algorithm for mechanical systems described in §2.4. The equations defining this

algorithm are

(q1, p1) = (q0, p0) + h∑j

(bjQj , bjPj), (2.5.1a)

(Qi, Pi) = (q0, p0) + h∑j

(aijQj , aijPj), (2.5.1b)

(Qj , Pj) = XH(Qj , Pj), (2.5.1c)

for some coefficients bj , bj , aij , aij satisfying Equation 2.4.2. These equations specify the push-

forward discrete Lagrange map for some discrete Lagrangian, as discussed in §2.4. We will assume

that there is an abelian group G that acts freely and properly on the configuration manifold Q, and

that the Lagrangian and the Hamiltonian functions are invariant under the lifted actions of G on TQ

and T ∗Q, respectively. Locally, Q ≈ G×S, where S = Q/G is the shape space. Let θ = (θ1, . . . , θr)

be local coordinates on G such that the group operation is addition, i.e., θ1 · θ2 = θ1 + θ2. (Since

the group is abelian, such coordinates can always be found.) Let x = (x1, . . . , xs) be coordinates

on S. In a local trivialization, (θ, x) are coordinates on Q. Let (θ, x, pθ, px) be canonical cotangent

bundle coordinates on T ∗Q, and (θ, x, θ, x) be canonical tangent bundle coordinates on TQ. It is

easy to show that in these canonical coordinates on T ∗Q, elements of the set J−1(µ) ⊂ T ∗Q are of

the form (θ, x, µ, px). Also, since the Hamiltonian H on T ∗Q is group invariant, H(θ, x, pθ, px) is

independent of θ. Note that here we are implicitly assuming that the vector space structure used to

define the SPRK method is that in which the group action is addition.

39

For the remainder of this section, we will adopt a local trivialization to express the SPRK method

in which the group action is addition. Rewriting the symplectic partitioned Runge–Kutta algorithm

in terms of this local trivialization gives

θ1 = θ0 + h∑j

bjΘj , (pθ)1 = (pθ)0 + h∑j

bj(Pθ)j , (2.5.2)

x1 = x0 + h∑

bjXj , (px)1 = (px)0 + h∑j

bj(Px)j , (2.5.3)

Θi = θ0 + h∑j

aijΘj , (Pθ)i = (pθ)0 + h∑j

aij(Pθ)j , (2.5.4)

Xi = x0 + h∑

aijXj , (Px)i = (px)0 + h∑j

aij(Px)j , (2.5.5)

and further,

Θj =∂H

∂pθ, (Pθ)j = −∂H

∂θ, (2.5.6)

Xj =∂H

∂px, (Px)j = −∂H

∂x. (2.5.7)

By group invariance, H does not depend on θ, and so ∂H/∂θ = 0. Thus (Pθ)j = 0, and therefore,

(pθ)1 = (Pθ)i = (pθ)0. Hence, if (q0, p0) ∈ J−1(µ), then (q1, p1) and (Qi, Pi) also lie on J−1(µ). (We

already know from the theory in the previous sections that the symplectic partitioned Runge–Kutta

algorithm preserves momentum; what we have verified here is that the intermediate points (Qi, Pi)

do not move off the momentum surface.)

If A is a connection on Q, it can be represented in local coordinates as

A(θ, x)(θ, x) = A(x)x+ θ.

Thus, the 1-form Aµ on Q is given by

Aµ(θ, x)(θ, x) = 〈µ,A(x)x+ θ〉 =[µ µA(x)

]θx

.Thus, Aµ(θ, x) = (θ, x, µ, µA(x)).

As we have seen in §2.2, there is a projection πµ : J−1(µ) → T ∗S. If αq ∈ J−1q (µ), (αq−Aµ(q)) ∈

T ∗qQ annihilates all vertical tangent vectors at q, and πµ(αq) is the element of T ∗xS determined by

(αq − Aµ(q)).

Suppose that in local coordinates, αq = (θ, x, µ, px). Then, (αq − Aµ(q)) = (θ, x, 0, px − µA(x)).

40

Thus, πµ(θ, x, µ, px) = (x, px − µA(x)). Therefore, Tπµ : TJ−1(µ) → T (T ∗S) is given by

Tπµ : (θ, x, 0, px) 7→ (x, px − µ∂A

∂xx).

In components, µA(x) can be represented as µaAai (x) (sum over the repeated index a is implicit),

and

µ∂A

∂xx = µa

∂Aai∂xj

xj .

Let (q, p) ∈ J−1(µ) and let (q, p) = XH(q, p). By Noether’s theorem, we have that (q, p) ∈

T(q,p)(J−1(µ)). In local coordinates,

(θ, x, 0, px) = XH(θ, x, µ, px).

Now, by the theory of cotangent bundle reduction (see §2.2),

Tπµ ·XH(q, p) = XHµ(πµ(q, p)),

i.e.,

(x, px − µ∂A

∂xx) = XHµ

(x, px − µA(x)).

If (q0, p0) ∈ J−1(µ), we have seen how (Qi, Pi) and (q1, p1) also lie in J−1(µ). Let

πµ(q0, p0) =: (x0, s0) = (x0, (px)0 − µA(x0)),

πµ(Qi, Pi) =: (Xi, Si) = (Xi, (Px)i − µA(Xi)),

πµ(q1, p1) =: (x1, s1) = (x1, (px)1 − µA(x1)).

Then,

(Xi, Si) := XHµ(Xi, Si) = (Xi, (Px)i − µ∂A

∂x(Xi)Xi). (2.5.8)

Remark 2.1. The Routh equations,

∂Rµ

∂x− d

dt

∂Rµ

∂x= ixβµ(x),

define a vector field on TS which is related to the vector field XHµby the reduced Legendre transform

FRµ. The equations

s =∂Rµ

∂x(x, x), (2.5.9)

41

and

s =∂Rµ

∂x(x, x)− ixβµ(x), (2.5.10)

can be used to solve for (x, s) in terms of (x, s), and thereby implicitly define the vector field XHµ.

Recall that

(px)1 = (px)0 + h∑j

bj(Px)j .

Adding and subtracting terms, this becomes

(px)1 − µA(x1) = (px)0 − µA(x0) + h∑j

bj

[(Px)j − µ

∂A

∂x(Xj)Xj

]

+

h∑j

(bjµ

∂A

∂x(Xj)Xj

)− (µA(x1)− µA(x0))

. (2.5.11)

This can be rewritten as

s1 = s0 + h∑j

bjSj +

h∑j

(bjµ

∂A

∂x(Xj)Xj

)− (µA(x1)− µA(x0))

. (2.5.12)

Similarly, it can be shown that

Si = s0 + h∑j

aijSj +

h∑j

(aijµ

∂A

∂x(Xj)Xj

)− (µA(Xi)− µA(x0))

. (2.5.13)

Putting the above equations together with the equations for x1 andXi, we get the following algorithm

on T ∗S:

x1 = x0 + h∑

bjXj , (2.5.14a)

s1 = s0 + h∑j

bjSj +

h∑j

(bjµ

∂A

∂x(Xj)Xj

)− (µA(x1)− µA(x0))

, (2.5.14b)

Xi = x0 + h∑

aijXj , (2.5.14c)

Si = s0 + h∑j

aijSj +

h∑j

(aijµ

∂A

∂x(Xj)Xj

)− (µA(Xi)− µA(x0))

, (2.5.14d)

Sj =∂Rµ

∂x(Xj , Xj), (2.5.14e)

Sj =∂Rµ

∂x(Xj , Xj)− iXj

βµ(Xj). (2.5.14f)

42

We shall refer to this system of equations as the reduced symplectic partitioned Runge–Kutta

(RSPRK) algorithm. Since we obtained this system by dropping the symplectic partitioned Runge–

Kutta algorithm from J−1(µ) to T ∗S, it follows that this algorithm preserves the reduced symplectic

form Ωµ = ΩS − π∗T∗S,Sβµ on T ∗S.

Since the SPRK algorithm is an integration algorithm for the Hamiltonian vector field XH on

T ∗Q, the RSPRK algorithm is an integration algorithm for the reduced Hamiltonian vector field

XHµon T ∗S. The relationship between cotangent bundle reduction and the reduction of the SPRK

algorithm can be represented by the following commutative diagram:

(J−1(µ), XH)

πµ

//___ (J−1(µ), SPRK)

πµ

(T ∗S,XHµ) //____ (T ∗S,RSPRK)

The dashed arrows here denote the corresponding discretization, as in Equation 2.3.32. We saw in

§2.4 that the SPRK algorithm can be obtained by pushing forward the DEL equations by the discrete

Legendre transform. By Lemma 2.9, this implies that the RSPRK algorithm can be obtained by

pushing forward the DR equations by the reduced discrete Legendre transform F = D2Ld − A2.

These relationships are shown in the following commutative diagram:

(J−1d (µ), DEL)

FLd //

πµ,d

(J−1(µ), SPRK)

πµ

(S × S,DR) F // (T ∗S,RSPRK)

2.6 Putting Everything Together

Let us now recapitulate some of the main results of the previous sections.

We saw in §2.2 that the relationship between Routh reduction and cotangent bundle reduction

can be represented by the following commutative diagram:

(J−1L (µ), EL) FL //

πµ,L

(J−1(µ), XH)

πµ

(TS,R) FRµ// (T ∗S,XHµ

)

We saw in §2.3.5 that if Ld approximates the Jacobi solution of the Hamilton–Jacobi equation, the

43

relationship between discrete and continuous Routh reduction is described by the following diagram:

(J−1L (µ), EL) //___

πµ,L

(J−1d (µ), DEL)

πµ,d

(TS,R) //_____ (S × S,DR)

The dashed arrows mean that the DEL equations are an integration algorithm for the EL equations,

and that the DR equations are an integration algorithm for the Routh equations.

If Ld is defined as in §2.4, we saw that the algorithm on T ∗Q obtained by pushing forward the

DEL equation using the discrete Legendre transform FLd is the symplectic partitioned Runge–Kutta

algorithm (Equation 2.4.1), which is an integration algorithm for XH . This is depicted as follows:

(J−1L (µ), EL) //___

FL

(J−1d (µ), DEL)

FLd

(J−1(µ), XH) //___ (J−1(µ), SPRK)

The SPRK algorithm on J−1(µ) ⊂ T ∗Q induces the RSPRK algorithm on J−1(µ)/G ≈ T ∗S. As we

saw in §2.5, this reduction process is related to cotangent bundle reduction and to discrete Routh

reduction as shown in the following diagram:

(J−1(µ), XH) //___

πµ

(J−1(µ), SPRK)

πµ

(J−1d (µ), DEL)

FLdoo

πµ,d

(T ∗S,XHµ) //____ (T ∗S,RSPRK) (S × S,DR)Foo

Putting all the above commutative diagrams together into one diagram, we obtain Figure 2.1.

2.7 Links with the Classical Routh Equations

The Routhian function Rµ that we have been using is not the same as the classical Routhian defined

by Routh [1877]. The classical Routhian, which we shall denote Rµc , is a function on TS that is

related to our Routhian by the equation

Rµc (x, x) = Rµ(x, x) + 〈µ,A(x)x〉.

Recall from §2.2 that the map A(x) : TxS → g is the restriction of the connection A to TxS. (TxS

is identified with the subspace TxS ×0 of TgG× TxS, which in turn is identified with TqQ.) Note

44

J−1(µ), XH//________

πµ

J−1(µ), SPRK

πµ

J−1L (µ), EL //_________

FL

??

πµ,L

J−1d (µ), DEL

FLd

??

πµ,d

T ∗S,XHµ_____ //____ T ∗S,RSPRK

TS,R //___________

FRµ

??S × S,DR

F

??

Figure 2.1: Complete commutative cube. Dashed arrows represent discretization from the contin-

uous systems on the left face to the discrete systems on the right face. Vertical arrows represent

reduction from the full systems on the top face to the reduced systems on the bottom face. Front

and back faces represent Lagrangian and Hamiltonian viewpoints, respectively.

that the map A(x) depends on our choice of local trivialization. Thus Rµc , too, depends on the

trivialization.

The classical Routh equations are

∂Rµc∂x

− d

dt

∂Rµc∂x

= 0. (2.7.1)

It can be verified (see, for example, Marsden and Ratiu [1999]) that these equations are equivalent

to the modern Routh equations (Equation 2.2.3), which we restate here:

∂Rµ

∂x− d

dt

∂Rµ

∂x= ixβµ(x). (2.7.2)

Thus the classical and the modern Routh equation define the same vector field X on TS.

To obtain dynamics on T ∗S, we could use the fiber derivative of either the modern Routhian

FRµ or that of the classical Routhian FRµc . In coordinates on TS and T ∗S, these fiber derivatives

are

FRµ : (x, x) ∈ TS 7→ (x,∂Rµ

∂x(x, x)) ∈ T ∗S, (2.7.3)

45

FRµc : (x, x) ∈ TS 7→ (x,∂Rµc∂x

(x, x)) = (x,∂Rµ

∂x(x, x) + µA(x)) ∈ T ∗S. (2.7.4)

Note that the map FRµc depends upon the trivialization.

We have seen in §2.2 that by pushing forward the dynamics on TS by FRµ, we obtain the vector

field XHµon T ∗S. Recall that the restriction of the Hamiltonian vector field XH to J−1(µ) is

πµ-related to XHµ , where πµ : J−1(µ) → T ∗S is the projection. Also recall that XHµ is Hamiltonian

with respect to the non-canonical symplectic structure Ωµ = ΩS − π∗T∗S,Sβµ on T ∗S.

If, on the other hand, we use FRµc to push forward the dynamics from TS to T ∗S, we obtain a

vector field (which we shall call XH′) that is Hamiltonian with respect to the canonical symplectic

structure ΩS on T ∗S.

Consider the following symplectic partitioned Runge–Kutta scheme for integrating XH′ :

x1 = x0 + hs∑j=1

bjXj , y1 = y0 + hs∑j=1

bj Yj , (2.7.5a)

Xi = x0 + hs∑j=1

aijXj , Yi = y0 + hs∑j=1

aij Yj , i = 1, . . . , s, (2.7.5b)

Yi =∂Rµc∂x

(Xi, Xi), Yi =∂Rµc∂x

(Xi, Xi), i = 1, . . . , s, (2.7.5c)

for some coefficients bj , bj , aij , aij satisfying Equation 2.4.2. It follows from that condition that this

scheme preserves the canonical symplectic structure ΩS . A particularly simple scheme of this form,

that is second-order, was developed independently by Sanyal et al. [2003].

A natural question to ask at this point is how the above integration scheme for the reduced

dynamics is related to the RSPRK scheme (Equation 2.5.14). To answer this question, consider the

map σ := FRµc (FRµ)−1 : T ∗S → T ∗S. In coordinates, σ : (x, s) 7→ (x, y) = (x, s + µA(x)). Note

that the σ transforms XHµto XH′ , i.e., XH′ = σ∗XHµ

. It can be verified that this map σ also

transforms the RSPRK scheme (Equation 2.5.14) to the above SPRK scheme for XH′ . Thus, these

two schemes for integrating the reduced dynamics are equivalent, and are related to each other by

a momentum shift.

Though the derivation for the SPRK scheme for XH′ (Equation 2.7.5) is shorter than the re-

duction process through which we obtained the RSPRK scheme (Equation 2.5.14), there are several

reasons to prefer the RSPRK scheme. Firstly, the classical Routhian Rµc and therefore the fiber

derivative FRµc and the vector field XH′ are dependent on the trivialization. Consequently, the

SPRK scheme for XH′ is non-intrinsic. On the other hand, as we saw in §2.5, the RSPRK scheme

(Equation 2.5.14) is derived by dropping the SPRK scheme (Equation 2.5.1) for XH onto the quo-

tient T ∗S = J−1(µ)/Gµ in a manner that is independent of the trivialization. (Though the equations

46

defining the RSPRK scheme have terms involving the map A(x), which is trivialization dependent,

the trivialization dependence “cancels out”, causing the overall scheme to be trivialization indepen-

dent.)

Secondly, since the vector field XH′ and the SPRK scheme (Equation 2.7.5) are not derived by

a reduction process, it is not possible to fit them in a natural way into a commutative diagram like

that depicted in Figure 2.1.

Furthermore, the classical theory of Routh reduction does not generalize to the case of non-

abelian symmetry groups, whereas the intrinsic, modern version does (see, for example, Marsden

and Scheurle [1993a,b], Jalnapurkar and Marsden [2000], and Marsden et al. [2000b]). Thus, to

develop numerical algorithms for the reduced dynamics of systems with non-abelian symmetry, one

would need to build on the intrinsic approach developed in this paper.

2.8 Forced and Constrained Systems

2.8.1 Constrained Coordinate Formalism

It is often desirable for computational reasons to realize the configuration space as a constraint

manifold Q in a containing space V.

Assume that the constraint manifold Q can be expressed as the preimage of a regular value of

a G-invariant constraint function, g : V → Rm. Then, g−1 (0) = Q ⊂ V is a constraint manifold of

codimension m.

On the constraint manifold Q, the discrete Hamilton’s variational principle states that

δ

n−1∑k=0

Ld (qk, qk+1) = 0

for all variations δq of q that vanish at the endpoints. By the Lagrange multiplier theorem, in the

containing space V , this is equivalent to the discrete Hamilton’s variational principle with

constraints,

δ

[n−1∑k=0

Ld (vk, vk+1) +n∑k=0

λTk g (vk)

]= 0,

for all variations δv of v that vanish at the endpoints.

As the variations are arbitrary and vanish at the endpoints, this is equivalent to the discrete

Euler–Lagrange equations with constraints,

D2Ld (vk−1, vk) +D1Ld (vk, vk+1) + λTkDg (vk) = 0,

g (vk) = 0.

47

In the case of higher-order discrete Lagrangians, one must be careful about the choice constrained

discrete Lagrangians. In particular, the internal sample points used in defining the constrained

discrete Lagrangian must lie on the constraint manifold Q. In practice, this corresponds to the

inclusion of Lagrange multiplier terms for each of the internal sample points in the variational

definition of the higher-order constrained discrete Lagrangian.

Consider the preshape space, U = V/G. As the constraint function g : V → Rm is G-invariant,

this induces the function g : U → Rm.

In addition, we have a G-invariant discrete Lagrangian, LVd : V × V → R, and the discrete

Lagrangian on Q×Q is simply the restriction, i.e., LQd = LVd∣∣Q×Q. The discrete momentum maps

are related by the following lemma.

Lemma 2.13. The discrete momentum map, JQd : Q×Q→ g∗, is obtained from JVd : V × V → g∗

by restriction, i.e., JQd = JVd∣∣Q×Q .

Proof. Since q0 ∈ Q ⊂ V , ξQ (q0) ∈ Tq0Q → Tq0V. Q is G-invariant, thus, the group orbits lie on Q,

and in particular, ξQ (q0) = ξV (q0) . The result then follows from the calculation:

JQd (q0, q1) · ξ = D1LQd (q0, q1) · ξQ (q0)

= D1LQd (q0, q1) · ξV (q0)

= D1LVd (q0, q1) · ξV (q0)

= JVd (q0, q1) · ξ.

Since the discrete momentum map on the constraint manifold is obtained by restriction, and in

our subsequent discussion, all the forms are evaluated on the constraint manifold, we shall abuse

notation and omit the superscripts denoting the spaces. We are thereby able to formulate the main

theorem of this section.

Theorem 2.14. Let x be a discrete curve on S, and let y be a discrete curve on U. Then, the

following are equivalent.

1. x solves the discrete Routh equations,

D2Ld (xk−1, xk) +D1Ld (xk, xk+1) = A2 (xk−1, xk) + A1 (xk, xk+1) .


δn−1∑k=0

Ld (xk, xk+1) =n−1∑k=0

A (xk, xk+1) · (δxk, δxk+1) ,

48


3. y solves the discrete (reduced) Routh equations with constraints,

D2Ld (yk−1, yk) +D1Ld (yk, yk+1) + λTkDg (yk) = A2 (yk−1, yk) + A1 (yk, yk+1) ,

g (yk) = 0.

4. y is a solution of the reduced discrete variational principle,

δ

[n−1∑k=0

Ld (yk, yk+1) +n∑k=0

λTk g (yk)

]=n−1∑k=0

A (yk, yk+1) · (δyk, δyk+1) ,

for all variations δy of y that vanish at the endpoints, and g (yk) = 0.

Proof. If q is a lift of x onto the µ-momentum surface, then the first two statements are equivalent

to the discrete Hamilton’s variational principle, which states that

δ

n−1∑k=0

Ld (qk, qk+1) = 0,

for all variations δq of q that vanish at the endpoints. By the Lagrange multiplier theorem, this is

equivalent to the discrete Hamilton’s variational principle with constraints,

δ

[n−1∑k=0


λTk g (vk)

]= 0,

for all variations δv of v that vanish at the endpoints.

As the variations are arbitrary and vanish at the endpoints, this is equivalent to the discrete

Euler–Lagrange equations with constraints,

D2Ld (vk−1, vk) +D1Ld (vk, vk+1) + λTkDg (vk) = 0,

g (vk) = 0.

Let v be a solution of the discrete Euler–Lagrange equations with constraints. Then,

δ

[n−1∑k=0


λTk g (vk)

]

=d

dε

∣∣∣∣ε=0

[n−1∑k=0

Ld (vkε, vk+1ε

) +n∑k=0

λTkεg (vkε

)

]

49

= D1Ld (v0, v1) · δv0 +n−1∑k=1

(D2Ld (vk−1, vk) +D1Ld (vk, vk+1)) · δvk

+D2Ld (vn−1, vn) · δvn +n∑k=0

g (vk)︸︷︷︸0

· δλk + λT0 Dg (v0) · δv0

+n−1∑k=1

(λTkDg (vk)

)· δvk + λTnDg (vn) · δvn

=(D1Ld (v0, v1) + λT0 Dg (v0)

)· δv0 +

(D2Ld (vn−1, vn) + λTnDg (vn)

)· δvn

+n−1∑k=1

[D2Ld (vk−1, vk) +D1Ld (vk, vk+1) + λTkDg (vk)

]︸︷︷︸0

=(D1Ld (v0, v1) + λT0 Dg (v0)

)· δv0 +


)· δvn.

Therefore, we have that v solves the discrete Euler–Lagrange equations with constraints if, and

only if,

d

dε

∣∣∣∣ε=0

[n−1∑k=0

Ld (vkε , vk+1ε) +n∑k=0

λTkεg (vkε)

]

=(D1Ld (v0, v1) + λT0 Dg (v0)

)· δv0 +


)· δvn,

for all variations, including those that do not vanish at the endpoints.

Let y be the projection of v, the solution of the DEL equations with constraints, onto the

preshape space V/G, and δy = ddε

∣∣ε=0

yε be a variation of y. By construction,

g (ykε) = g (vkε) .

The terms λT0 Dg (v0) and λTnDg (vn) correspond to forces of constraint, and are therefore normal

to the constraint manifold. Since the constraint manifold Q is G-invariant, the group orbits lie on

the constraint manifold. As a consequence, the forces of constraint annihilate vertical variations,

implying that

λT0 Dg (v0) · ver δv0 = 0,

λTnDg (vn) · ver δvn = 0.

From which we conclude,

d

dε

∣∣∣∣ε=0

[n−1∑k=0

Ld (vkε , vk+1ε) +n∑k=0

λTkεg (vkε)

]

50

=(D1Ld (v0, v1) + λT0 Dg (v0)

)· δv0 +


)· δvn

=(D1Ld (v0, v1) + λT0 Dg (v0)

)· hor δv0 +


)· hor δvn

− Aµ (v0) · δv0 + Aµ (vn) · δvn

=(D1Ld (v0, v1) + λT0 Dg (v0)

)· hor δv0 +


)· hor δvn

+n−1∑k=0

A (vk, vk+1) · (δvk, δvk+1) ,

where we used Equation 2.3.19 for the second to last equality. Then,

(D1Ld (v0, v1) + λT0 Dg (v0)

)· hor δv0 +


)· hor δvn

=d

dε

∣∣∣∣ε=0

[n−1∑k=0

Ld (vkε, vk+1ε

) +n∑k=0

λTkεg (vkε

)

]−n−1∑k=0

A (vk, vk+1) · (δvk, δvk+1) .

From Lemma 2.2, and the fact that g (ykε) = g (vkε

), this can be rewritten in terms of the reduced

quantities,

(D1Ld (v0, v1) + λT0 Dg (v0)

)· hor δv0 +


)· hor δvn

=d

dε

∣∣∣∣ε=0

[n−1∑k=0

Ld (ykε, yk+1ε

) +n∑k=0

λTkεg (ykε

)

]−n−1∑k=0

A (yk, yk+1) · (δyk, δyk+1) .

If the variations δy vanishes at the endpoints, i.e., δy0 = δyn = 0, then hor δv0 = hor δvn = 0,

and therefore

δ

[n−1∑k=0


λTk g (yk)

]=n−1∑k=0

A (yk, yk+1) · (δyk, δyk+1) ,


Since the variations are arbitrary and vanish at the endpoints, this is equivalent to the Discrete

Routh equations with constraints,

D2Ld (yk−1, yk) +D1Ld (yk, yk+1) + λTkDg (yk) = A2 (yk−1, yk) + A1 (yk, yk+1) ,

g (yk) = 0.

Conversely, if y satisfies the reduced variational principle, and v is its lift onto the µ-momentum

surface, then a construction analogous to the derivation of the discrete Routh equations shows that

v satisfies the discrete Hamilton’s variational principle with constraints.

51

2.8.2 Routh Reduction with Forcing

Mechanical systems with external forcing are governed by the Lagrange–d’Alembert variational

principle,

δ

∫L (q (t) , q (t)) dt+

∫F (q (t) , q (t)) · δqdt = 0.

We define the discrete Lagrange–d’Alembert principle (Kane et al. [2000]) to be

δn−1∑k=0

Ld (qk, qk+1) +n−1∑k=0

Fd (qk, qk+1) · (δqk, δqk+1) = 0,

for all variations δq of q that vanish at the endpoints. Fd is a 1-form on Q×Q, and approximates the

impulse integral between the points qk and qk+1, just as the discrete Lagrangian Ld approximates

the action integral. We define the 1-forms F+d and F−d on Q×Q and the maps F 1

d , F2d : Q×Q→ T ∗Q

by the relations

F+d (q0, q1) · (δq0, δq1) = F 2

d (q0, q1) · δq1 = Fd (q0, q1) · (0, δq1) ,

F−d (q0, q1) · (δq0, δq1) = F 1d (q0, q1) · δq0 = Fd (q0, q1) · (δq0, 0) .

The discrete Lagrange–d’Alembert principle may then be rewritten as

δn−1∑k=0

Ld (qk, qk+1) +n−1∑k=0

[F 1d (qk, qk+1) · δqk + F 2

d (qk, qk+1) · δqk+1

]= 0,

for all variations δq of q that vanish at the endpoints. This is equivalent to the forced discrete

Euler–Lagrange equations,

D2Ld (qk−1, qk) +D1Ld (qk, qk+1) + F 1d (qk, qk+1) + F 2

d (qk−1, qk) = 0.

As we are concerned with mechanical systems with symmetry, we shall restrict our discussion

to discrete forces that are invariant under the diagonal action of G on Q×Q. In particular, for all

ξ ∈ g, and all variations (δq0, δq1) of (q0, q1) ,

Fd (exp (tξ) q0, exp (tξ) q1) · (δq0, δq1) = Fd (q0, q1) · (δq0, δq1) .

Since the Routh reduction technique requires that the momentum map be conserved, we shall

further restrict our discussion to G-invariant forcing that satisfies the discrete Noether theorem.

This constrains our choice of forcing, as the following lemma illustrates.

Lemma 2.15. Let q be a discrete curve on Q that solves the forced discrete Euler–Lagrange equa-

52

tions. Then, the discrete Noether theorem is satisfied if, and only if,

(F 2d (qk−1, qk) + F 1

d (qk, qk+1))· ver δqk = 0.

Proof. Given ξ ∈ g, consider the ξ-component of Jd, given by

Jξd (q0, q1) = 〈Jd (q0, q1) , ξ〉 .

We compute the evolution of Jξd along the flow of the forced discrete Euler–Lagrange equations:

Jξd (q1, q2)− Jξd (q0, q1)

= Jd (q1, q2) · ξ − Jd (q0, q1) · ξ

= −D1Ld (q1, q2) · ξQ (q1)−D2Ld (q0, q1) · ξQ (q1)

= −D1Ld (q1, q2) · ξQ (q1)−D2Ld (q0, q1) · ξQ (q1)

+[D2Ld (q0, q1) +D1Ld (q1, q2) + F 2

d (q0, q1) + F 1d (q1, q2)

]︸︷︷︸0

· ξQ (q1)

=[F 2d (q0, q1) + F 1

d (q1, q2)]· ξQ (q1) .

Since Jd : Q × Q → g∗, the discrete Noether theorem is satisfied if, and only if, Jξd (q1, q2) −

Jξd (q0, q1) = 0, for all ξ ∈ g. As the vertical space verq1 is given by

verq1 = ξQ (q1) | ξ ∈ g ,

this is equivalent to F 2d (q0, q1) + F 1

d (q1, q2) vanishing on all vertical vectors.

For the rest of our discussion, we shall specialize to the case whereby F 2d (q0, q1) and F 1

d (q1, q2)

individually vanish on vertical vectors, which is a sufficient condition for momentum conservation.

The discrete forcing term Fd is an invariant 1-form under the diagonal action of G on Q × Q,

and vanishes on vertical vectors. By restricting Fd to J−1d (µ), it drops to Fd : S × S → T ∗S × T ∗S.

In this context, we may formulate a discrete Routh reduction theory for the discrete Lagrange–

d’Alembert principle.

Theorem 2.16. Let x be a discrete curve on S, and let q be a discrete curve on Q with momentum

µ that is obtained by lifting x. Then, the following are equivalent.

1. q solves the forced discrete Euler–Lagrange equations,

D2Ld (qk−1, qk) +D1Ld (qk, qk+1) + F 2d (qk−1, qk) + F 1

d (qk, qk+1) = 0.

53

2. q is a solution of the discrete Lagrange–d’Alembert variational principle,

δn−1∑k=0

Ld (qk, qk+1) +n−1∑k=0

[F 1d (qk, qk+1) · δqk + F 2


]= 0,


3. x solves the Discrete Routh equations with forcing,

D2Ld (xk−1, xk) +D1Ld (xk, xk+1) + F 2d (xk−1, xk) + F 1

d (xk, xk+1)

= A2 (xk−1, xk) + A1 (xk, xk+1) .


δn−1∑k=0

Ld (xk, xk+1) +n−1∑k=0

[F 1d (xk, xk+1) · δxk + F 2

d (xk, xk+1) · δxk+1

]=n−1∑k=0

A (xk, xk+1) · (δxk, δxk+1) ,


Proof. We begin with the discrete Lagrange–d’Alembert variational principle,

δ

n−1∑k=0

Ld (qk, qk+1) +n−1∑k=0

[F 1d (qk, qk+1) · δqk + F 2


]= 0,


Since the variations are arbitrary and vanish at the endpoints, this is equivalent to the forced

discrete Euler–Lagrange equations,


d (qk, qk+1) = 0.

Let q be a solution of the forced discrete Euler–Lagrange equations, then,

δn−1∑k=0

Ld (qk, qk+1) +n−1∑k=0

[F 1d (qk, qk+1) · δqk + F 2


]=

d

dε

∣∣∣∣ε=0

n−1∑k=0

Ld (qkε , qk+1ε) +n−1∑k=0

[F 1d (qk, qk+1) · δqk + F 2


]= D1Ld (q0, q1) · δq0 +

n−1∑k=1

(D2Ld (qk−1, qk) +D1Ld (qk, qk+1)) · δqk

54

+D2Ld (qn−1, qn) · δqn + F 1d (q0, q1) · δq0

+n−1∑k=1

(F 1d (qk, qk+1) + F 2

d (qk−1, qk))· δqk + F 2

d (qn−1, qn) · δqn

=(D1Ld (q0, q1) + F 1

d (q0, q1))· δq0 +

(D2Ld (qn−1, qn) + F 2

d (qn−1, qn))· δqn

+n−1∑k=1

[D2Ld (qk−1, qk) +D1Ld (qk, qk+1) + F 2

d (qk−1, qk) + F 1d (qk, qk+1)

]︸︷︷︸0

· δqk

=(D1Ld (q0, q1) + F 1

d (q0, q1))· δq0 +

(D2Ld (qn−1, qn) + F 2

d (qn−1, qn))· δqn.

Conversely, for an arbitrary discrete curve q and an arbitrary variation δq, the final equality

only holds if q satisfies


d (qk, qk+1) = 0,

which is the forced DEL equation.

Therefore, we have that q solves the forced discrete Euler–Lagrange equations if, and only if,

d

dε

∣∣∣∣ε=0

n−1∑k=0

Ld (qkε , qk+1ε) +n−1∑k=0

[F 1d (qk, qk+1) · δqk + F 2


]=(D1Ld (q0, q1) + F 1

d (q0, q1))· δq0 +

(D2Ld (qn−1, qn) + F 2

d (qn−1, qn))· δqn,

for all variations, including those that do not vanish at the endpoints.

Let x be the projection of q, the solution of the forced DEL equations, onto the shape space

S, and δx = ddε

∣∣ε=0

xε be a variation of x. Since (qk, qk+1) is on the µ-momentum surface, and

(δqk, δqk+1) is tangent to the momentum surface, we have by the construction of Fd the following

relations

F 1d (xk, xk+1) · δxk + F 2


= Fd (xk, xk+1) · (δxk, 0) + Fd (xk, xk+1) · (0, δxk+1)

= Fd (xk, xk+1) · (δxk, δxk+1)

= Fd (qk, qk+1) · (δqk, δqk+1)

= Fd (qk, qk+1) · (δqk, 0) + Fd (qk, qk+1) · (0, δqk+1)

= F 1d (qk, qk+1) · δqk + F 2

d (qk, qk+1) · δqk+1.

This allows us to rewrite the sum over discrete forces in the discrete Lagrange–d’Alembert prin-

55

ciple in terms of a sum over the reduced discrete forces,

d

dε

∣∣∣∣ε=0

n−1∑k=0

Ld (qkε, qk+1ε

) +n−1∑k=0

[F 1d (xk, xk+1) · δxk + F 2


]=(D1Ld (q0, q1) + F 1

d (q0, q1))· δq0 +

(D2Ld (qn−1, qn) + F 2

d (qn−1, qn))· δqn.

Splitting the variations into horizontal and vertical components, and using the assumption that

the discrete forces vanish on vertical vectors, we have

(D1Ld (q0, q1) + F 1

d (q0, q1))· δq0 +

(D2Ld (qn−1, qn) + F 2

d (qn−1, qn))· δqn

=(D1Ld (q0, q1) + F 1

d (q0, q1))· (ver δq0 + hor δq0)

+(D2Ld (qn−1, qn) + F 2

d (qn−1, qn))· (ver δqn + hor δqn)

=(D1Ld (q0, q1) + F 1

d (q0, q1))· hor δq0 +

(D2Ld (qn−1, qn) + F 2

d (qn−1, qn))· hor δqn

− Aµ (q0) · δq0 + Aµ (qn) · δqn

=(D1Ld (q0, q1) + F 1

d (q0, q1))· hor δq0 +

(D2Ld (qn−1, qn) + F 2


+n−1∑k=0

A (qk, qk+1) · (δqk, δqk+1) ,

where, as before, we used Equation 2.3.19 for the second to last equality. Then,

(D1Ld (q0, q1) + F 1

d (q0, q1))· hor δq0 +

(D2Ld (qn−1, qn) + F 2


=d

dε

∣∣∣∣ε=0

n−1∑k=0

Ld (qkε, qk+1ε

) +n−1∑k=0

[F 1d (xk, xk+1) · δxk + F 2


]−n−1∑k=0

A (qk, qk+1) · (δqk, δqk+1) .

If the variations δx vanishes at the endpoints, i.e., δx0 = δxn = 0, then hor δq0 = hor δqn = 0,

and therefore,

δn−1∑k=0

Ld (xkε, xk+1ε

) +n−1∑k=0

[F 1d (xk, xk+1) · δxk + F 2


]=n−1∑k=0

A (xk, xk+1) · (δxk, δxk+1) ,


Since the variations are arbitrary and vanish at the endpoints, this is equivalent to the Discrete

56

Routh equations with forcing,

D2Ld (xk−1, xk) +D1Ld (xk, xk+1) + F 2d (xk−1, xk) + F 1

d (xk, xk+1)

= A2 (xk−1, xk) + A1 (xk, xk+1) .

Conversely, if x satisfies the reduced variational principle, and q is its lift onto the µ-momentum

surface, then a construction analogous to the derivation of the discrete Routh equations show that

q satisfies the discrete Lagrange–d’Alembert principle.

2.8.3 Routh Reduction with Constraints and Forcing

By applying the techniques of the previous sections, we may synthesize the formalisms involving

constraints and forcing. We shall state, without proof, the relevant equations in the following

theorem.

Theorem 2.17. Let x be a discrete curve on S, and let q be a discrete curve on Q with momentum

µ that is obtained by lifting x. Let y be a discrete curve on U obtained from x by the inclusion

S = g−1(0) → U , and let v be a discrete curve on V with momentum µ that is obtained by lifting

y. Then, the following are equivalent.

1. v solves the forced discrete Euler–Lagrange equations with constraints,

D2Ld (vk−1, vk) +D1Ld (vk, vk+1) + F 2d (vk−1, vk) + F 1

d (vk, vk+1) + λTkDg (vk) = 0,

g (vk) = 0.

2. v is a solution of the discrete Lagrange–d’Alembert variational principle with constraints,

δ

[n−1∑k=0


λTk g (vk)

]

+n−1∑k=0

[F 1d (vk, vk+1) · δvk + F 2

d (vk, vk+1) · δvk+1

]= 0,

for all variations δv of v that vanish at the endpoints, and g (vk) = 0.

3. y solves the Discrete Routh equations with forcing and constraints,

D2Ld (yk−1, yk) +D1Ld (yk, yk+1) + F 2d (yk−1, yk) + F 1

d (yk, yk+1) + λTkDg (yk)

= A2 (yk−1, yk) + A1 (yk, yk+1) ,

g (yk) = 0.

57

4. y is a solution of the reduced variational principle,

δ

[n−1∑k=0


λTk g (yk)

]

+n−1∑k=0

[F 1d (yk, yk+1) · δyk + F 2

d (yk, yk+1) · δyk+1

]=n−1∑k=0

A (yk, yk+1) · (δyk, δyk+1) ,


2.9 Example: J2 Satellite Dynamics

2.9.1 Configuration Space and Lagrangian

An illustrative and important example of a system with an abelian symmetry group is that of a single

satellite in orbit about an oblate Earth. The general aspects and background for this problem are

discussed in Prussing and Conway [1993], and some interesting aspects of the geometry underlying

it are discussed in Chang and Marsden [2003].

The configuration manifold Q is R3, and the Lagrangian for the system has the form, kinetic

minus potential energy,

L(q, (q)) =12Ms‖q‖2 −MsV (q),

where Ms is the mass of the satellite and V : R3 → R is the gravitational potential due to the Earth,

truncated at the first term in the expansion in the ellipticity,

V (q) =GMe

‖q‖+GMeR

2eJ2

‖q‖3

(32

(q3)2

‖q‖2− 1

2

).

Here, G is the gravitational constant, Me is the mass of the Earth, Re is the radius of the Earth, J2 is

a small non-dimensional parameter describing the degree of ellipticity, and q3 is the third component

of q.

We will now assume that we are working in non-dimensional coordinates, so that

L(q, q) =12‖q‖2 −

[1‖q‖

+J2

‖q‖3

(32

(q3)2

‖q‖2− 1

2

)]. (2.9.1)

This corresponds to choosing space and time coordinates in which the radius of the Earth is 1 and

the period of orbit at zero altitude is 2π when J2 = 0 (spherical Earth).

58

2.9.2 Symmetry Action

The symmetry of interest to us is that of rotation about the vertical (q3) axis, so the symmetry

group is the unit circle S1. Using cylindrical coordinates, q = (r, θ, z), for the configuration, the

symmetry action is φ : (r, θ, z) 7→ (r, θ + φ, z). Since ‖q‖, ‖q‖, and q3 = z are all invariant under

this transformation, so too is the Lagrangian.

This action is clearly not free on all of Q = R3, as the z-axis is invariant for all group elements.

This is not a serious obstacle, however, as the lifted action is free on T (Q\(0, 0, 0)) and this is enough

to permit the application of the intrinsic Routh reduction theory outlined in §2.2. Alternatively, one

can simply take Q = R3 \ (0, 0, z) | z ∈ R and then the theory literally applies.

The shape space, S = Q/G, is the half-plane S = R+ × R and we will take coordinates (r, z) on

S. In doing so, we are implicitly defining a global diffeomorphism S ×G→ Q given by ((r, z), θ) 7→

(r, θ, z).

The Lie algebra g for G = S1 is the real line g = R, and we will identify the dual with the real

line itself, g∗ ∼= R. For a Lie algebra element ξ ∈ g, the corresponding infinitesimal generator is

given by

ξQ : (r, θ, z) 7→ ((r, θ, z), (0, ξ, 0)).

Recall that the Lagrange momentum map, JL : TQ→ g∗, is defined by

JL(vq) · ξ = 〈FL(vq), ξQ(q)〉,

so we have

JL((r, θ, z), (r, θ, z)) · ξ =⟨(r, r2θ, z), (0, ξ, 0)

⟩= r2θξ,

and

JL((r, θ, z), (r, θ, z)) = r2θ.

This momentum map is simply the vertical component of the standard angular momentum.

Consider the Euclidean metric on R3, which corresponds to the kinetic energy norm in the

Lagrangian. From this metric we define the mechanical connection, A : TQ → g, which is given

by A((r, θ, z), (r, θ, z)) = θ. The 1-form Aµ on Q is thus given by Aµ = µdθ. Taking the exterior

derivative of this expression gives dAµ = µd2θ = 0, and so the reduced 2-form is βµ = 0.

59

2.9.3 Equations of Motion

Computing the Euler–Lagrange equations for the Lagrangian (Equation 2.9.1) gives the equations

of motion,

q = −∇q[

1‖q‖

+J2

‖q‖3

(32

(q3)2

‖q‖2− 1

2

)].

To calculate the reduced equations, we begin by calculating the Routhian,

Rµ(r, θ, z, r, θ, z) = L(r, θ, z, r, θ, z)− Aµ(r, θ, z) · (r, θ, z)

=12‖(r, θ, z)‖2 −

[1r

+J2

r3

(32z2

r2− 1

2

)]− µθ.

We now choose a fixed value µ of the momentum and restrict ourselves to the space J−1L (µ), on

which θ = µ. The reduced Routhian, Rµ : TS → R, is the restricted Routhian dropped to the

tangent bundle of the shape space. In coordinates, this is

Rµ(r, z, r, z) =12‖(r, z)‖2 −

[1r

+J2

r3

(32z2

r2− 1

2

)]− 1

2µ2.

Recalling that βµ = 0, the Routh equations (Equation 2.2.3) can now be evaluated to give

(r, z) = −∇(r,z)

[1r

+J2

r3

(32z2

r2− 1

2

)],

which describes the motion on the shape space.

To recover the unreduced Euler–Lagrange equations from the Routh equations, one uses the

procedure of reconstruction. This is covered in detail in Marsden et al. [1990], Marsden [1992] and

Marsden et al. [2000b].

2.9.4 Discrete Lagrangian System

We now discretize this system with the discrete Lagrangian used in Theorem 2.12. Recall that

the push-forward discrete Lagrange map associated with this discrete Lagrangian is a symplectic

partitioned Runge–Kutta method with coefficients given by Equation 2.4.3.

Given a point (q0, q1) ∈ Q × Q we will take (q0, p0) and (q1, p1) to be the associated discrete

Legendre transforms. As the discrete momentum map is the pull-back of the canonical momentum

map, we have that

JLd(q0, q1) = (pθ)0 = (pθ)1.

Take a fixed momentum map value µ and restrict Ld to the set J−1Ld

(µ). Dropping this to S×S now

gives the reduced discrete Lagrangian, Ld : S × S → R.

60

As discussed in §2.5, the fact that we have taken coordinates in which the group action is

addition in θ means that the push-forward discrete Lagrange map associated with the reduced

discrete Lagrangian is the reduced method given by Equation 2.5.14. In fact, as the mechanical

connection has A(r, z) = 0 and βµ = 0, the push-forward discrete Lagrange map is exactly a

partitioned Runge–Kutta method with Hamiltonian equal to the reduced Routhian. As we saw in

§2.7, these are generically related by a momentum shift, rather than being equal.

Given a trajectory of the reduced discrete system, we can reconstruct a trajectory of the unre-

duced discrete system by solving for the θ component of Equation 2.5.1. Correspondingly, a trajec-

tory of the unreduced discrete system can be projected onto the shape space to give a trajectory of

the reduced discrete system.

2.9.5 Example Trajectories

Solutions of the Spherical Earth System. Consider initially the system with J2 = 0. This

corresponds to the case of a spherical Earth, and so the equations reduce to the standard Kepler

problem. As this is an integrable system, the trajectories consist of periodic orbits.

A slightly inclined circular trajectory is shown in Figure 2.2, in both the unreduced and reduced

pictures. Note that the graph of the reduced trajectory is a quadratic, as ‖q‖ =√r2 + z2 is a

constant.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

y

0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

r

z

Figure 2.2: Unreduced (left) and reduced (right) views of an inclined elliptic trajectory for the

continuous time system with J2 = 0 (spherical Earth).

We will now investigate the effect of two different perturbations to the system, one due to taking

non-zero J2 and the other due to the numerical discretization.

61

The J2 Effect. Taking J2 = 0.05 (which is close the actual value for the Earth), the system

becomes near-integrable and experiences breakup of the KAM tori. This can be seen in Figure 2.3,

where the same initial condition is used as in Figure 2.2.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

y

0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

r

zFigure 2.3: Unreduced (left) and reduced (right) views of an inclined elliptic trajectory for the

continuous time system with J2 = 0.05. Observe that the non-spherical terms introduce precession

of the near-elliptic orbit in the symmetry direction.

Due to the fact that the reduced trajectory is no longer a simple curve, there is a geometric-

phase-like effect which causes precession of the orbit. This precession can be seen in the thickening

of the unreduced trajectory.

Solutions of the Discrete System for a Spherical Earth. We now consider the discrete

system with J2 = 0, for the second-order Gauss–Legendre discrete Lagrangian with timestep of

h = 0.3. The trajectory with the same initial condition as above is given in Figure 2.4.

As can be seen from the reduced trajectory, the discretization has caused a similar breakup of

the periodic orbit as was produced by the non-zero J2. The effect of this is to, once again, induce

precession of the orbit in the unreduced trajectory, in a way which is difficult to distinguish from

the perturbation above due to non-zero J2 when only the unreduced picture is considered. If the

reduced pictures are consulted, however, then it is immediately clear that the system is much closer

to the continuous time system with J2 = 0 than to the system with non-zero J2.

Solutions of the Discrete System with J2 Effect. Finally, we consider the discrete system

with non-zero J2 = 0.05. The resulting trajectory is shown in Figure 2.5, and, clearly, it is not easy

to determine from the unreduced picture whether the precession is due to the J2 perturbation, the

discretization, or some combination of the two.

62

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

y

0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

r

zFigure 2.4: Unreduced (left) and reduced (right) views of an inclined trajectory of the discrete

system with step-size h = 0.3 and J2 = 0. The initial condition is the same as that used in Figure

2.2. The numerically introduced precession means that the unreduced picture looks similar to that

of Figure 2.3 with non-zero J2, whereas, by considering the reduced picture we can see the correct

resemblance to the J2 = 0 case of Figure 2.2.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

y

0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

r

z

Figure 2.5: Unreduced (left) and reduced (right) views of an inclined trajectory of the discrete

system with step-size h = 0.3 and J2 = 0.05. The initial condition is the same as that used in

Figure 2.3. The unreduced picture is similar to that of both Figures 2.3 and 2.4. By considering the

reduced picture, we obtain the correct resemblance to Figure 2.3.

63

Taking the reduced trajectories, however, immediately shows that this discrete time system

is structurally much closer to the non-zero J2 system than to the original J2 = 0 system. This

confusion arises because both the J2 term and the discretization introduce perturbations which act

in the symmetry direction.

While this system is sufficiently simple that one can run simulations with such small timesteps

that the discretization artifacts become negligible, this is not possible in general. This example

demonstrates how knowledge of the geometry of the system can be important in understanding the

discretization process, and how this can give insight into the behavior of numerical simulations. In

particular, understanding how the discretization interacts with the symmetry action is extremely

important.

2.9.6 Coordinate Systems

In this example, we have chosen cylindrical coordinates, thus making the group action addition in

θ. One can always do this, as an abelian Lie group is isomorphic to a product of copies of R and S1,

but it may sometimes be preferable to work in coordinates in which the group action is not addition.

For example, cartesian coordinates in the present example.

It may be easier, both in terms of computational expense, and in the simplicity of expressions,

if we adopt a coordinate system in which the group action is not addition. We can still apply the

Discrete Routh equations to obtain an integration scheme on S×S. The push-forward of this under

F yields an integration scheme on T ∗S. The trajectories on the shape space that we obtain in this

manner could be different from those we would get with the RSPRK method. However, in both

cases we would have conservation of symplectic structure, momentum, and the order of accuracy

would be the same. One could choose whichever approach is cheaper and easier.

2.10 Example: Double Spherical Pendulum

2.10.1 Configuration Space and Lagrangian

We consider the example of the double spherical pendulum which has a non-trivial magnetic term

and constraints. The configuration manifold Q is S2 × S2, and the embedding linear space V is

R3 × R3.

The position vectors of each pendulum with respect to their pivot point are denoted by q1 and

q2, as illustrated in Figure 2.6. These vectors are constrained to have lengths l1 and l2, respectively,

and the pendula masses are denoted by m1 and m2.

64

q1

q2

m1

m2

l1

l2

k

FF __

>>

>>>>

>>>>

>>>>

>>>>

>

OO

Figure 2.6: Double spherical pendulum.

The Lagrangian for the system has the form, kinetic minus potential energy,

L(q1,q2, q1, q2) =12m1‖q1‖2 +

12m2‖q1 + q2‖2 −m1gq1 · k−m2g(q1 + q2) · k,

where g is the gravitational constant, and k is the unit vector in the z direction. The constraint

function, c : V → R2, is given by

c(q1,q2) = (‖q1‖ − l1, ‖q2‖ − l2).

Using cylindrical coordinates, qi = (ri, θi, zi), for the configuration, we can express the Lagrangian

as

L(q, q) =12m1

(r21 + r21 θ

21 + z2

1

)+

12m2

r21 + r21 θ

21 + r22 + r22 θ

22

+2(r1r2 + r1r2θ1θ2

)cosϕ+ 2

(r1r2θ1 − r2r1θ2

)sinϕ+ (z1 + z2)

2

−m1gz1 −m2g (z1 + z2) ,

where ϕ = θ2 − θ1. Furthermore, we can automatically satisfy the constraints by performing the

following substitutions,

zi =√l2i − r2i , zi = − riri√

l2i − r2i.

65

2.10.2 Symmetry Action

The symmetry of interest to us is the simultaneous rotation of the two pendula about vertical (z)

axis, so the symmetry group is the unit circle S1. Using cylindrical coordinates, qi = (ri, θi, zi), for

the configuration, the symmetry action is φ : (ri, θi, zi) 7→ (ri, θi+φ, zi). Since ‖qi‖, ‖qi‖, ‖q1+ q2‖,

and qi · k are all invariant under this transformation, so too is the Lagrangian.

This action is clearly not free on all of V = R3 × R3, as the z-axis is invariant for all group

elements. However, this does not pose a problem computationally, as long as the trajectories do

not pass through the downward hanging configuration, corresponding to r1 = r2 = 0. To treat the

downward handing configuration properly, we would need to develop a discrete Lagrangian analogue

of the continuous theory of singular reduction described in Ortega and Ratiu [2001].

The Lie algebra g for G = S1 is the real line g = R, and we will identify the dual with the real

line itself g∗ ∼= R. For a Lie algebra element ξ ∈ g, the corresponding infinitesimal generator is given

by

ξQ : (r1, θ1, z1, r2, θ2, z2) 7→ ((r1, θ1, z1, r2, θ2, z2), (0, ξ, 0, 0, ξ, 0)).

Recall that the Lagrange momentum map JL : TQ→ g∗ is defined by

JL(vq) · ξ = 〈FL(vq), ξQ(q)〉,

so we have

JL((r1, θ1, z1, r2, θ2, z2), (r1, θ1, z1, r2, θ2, z2)) · ξ

=⟨(m1r1 +m2

[r1 + r2 cosϕ− r2θ2 sinϕ

],

m1r21 θ1 +m2

[r21 θ1 + r1r2θ2 cosϕ+ r1r2 sinϕ

],

m1z1 +m2 [z1 + z2] ,m2

[r2 + r1 cosϕ+ r1θ1 sinϕ

],

m2

[r22 θ2 + r1r2θ1 cosϕ− r2r1 sinϕ

],m2 [z1 + z2]

), (0, ξ, 0, 0, ξ, 0)

⟩=((m1 +m2) r21 θ1 +m2r

22 θ2 +m2r1r2

(θ1 + θ2

)cosϕ+ (r1r2 − r2r1) sinϕ

)ξ,

and

JL((r1, θ1, z1, r2, θ2, z2), (r1, θ1, z1, r2, θ2, z2))

= (m1 +m2) r21 θ1 +m2r22 θ2 +m2r1r2

(θ1 + θ2

)cosϕ+ (r1r2 − r2r1) sinϕ.

This momentum map is simply the vertical component of the standard angular momentum.

66

The locked inertia tensor is given by Marsden [1992],

I(q1q2) = m1‖q⊥1 ‖2 +m2‖(q1 + q2)⊥‖2

= m1r21 +m2(r21 + r22 + 2r1r2 cosϕ).

Furthermore, the mechanical connection is given by

α(q1,q2, q1, q2) = I(q1,q2)−1JL(q1,q2, q1, q2)

=(m1 +m2) r21 θ1 +m2r

22 θ2 +m2r1r2

(θ1 + θ2

)cosϕ+ (r1r2 − r2r1) sinϕ

m1r21 +m2(r21 + r22 + 2r1r2 cosϕ).

As a 1-form, it is given by

α(q1,q2) =1

m1r21 +m2(r21 + r22 + 2r1r2 cosϕ)

×[(m1 +m2) r21dθ1 +m2r

22dθ2 +m2r1r2 (dθ1 + dθ2) cosϕ

+(r1dr2 − r2dr1) sinϕ] .

The µ-component of the mechanical connection is given by

αµ(q1,q2) =µ

m1r21 +m2(r21 + r22 + 2r1r2 cosϕ)

×[

(m1 +m2) r21 +m2r1r2 cosϕ]dθ1 +

[m2r

22 +m2r1r2 cosϕ

]dθ2.

Taking the exterior derivative of this 1-form yields a non-trivial magnetic term on the reduced space,

dαµ =µ

[m1r21 +m2 (r21 + r22 + 2r1r2 cosϕ)]2

× m2r2[2 (m1 +m2) r1r2 +

(m1r

21 +m2(r21 + r22)

)cosϕ

]dr1 ∧ dθ1

−m2r1[2 (m1 +m2) r1r2 +

(m1r

21 +m2(r21 + r22)

)cosϕ

]dr2 ∧ dθ1

+m2r1r2 sinϕ[m1r21 +m2(r21 − r22)]dθ2 ∧ dθ1

−m2r2[2 (m1 +m2) r1r2 +

(m1r

21 +m2(r21 + r22)

)cosϕ

]dr1 ∧ dθ2

+m2r1[2 (m1 +m2) r1r2 +

(m1r

21 +m2(r21 + r22)

)cosϕ

]dr2 ∧ dθ2

+m2r1r2 sinϕ[m1r21 +m2(r21 − r22)]dθ1 ∧ dθ2.

67

This 2-form drops to the quotient space to yield

βµ =µ

[m1r21 +m2 (r21 + r22 + 2r1r2 cosϕ)]2

× m2r2[2 (m1 +m2) r1r2 +

(m1r

21 +m2(r21 + r22)

)cosϕ

]dϕ ∧ dr1

−m2r1[2 (m1 +m2) r1r2 +

(m1r

21 +m2(r21 + r22)

)cosϕ

]dϕ ∧ dr2

=µm2

[2 (m1 +m2) r1r2 +

(m1r

21 +m2(r21 + r22)

)cosϕ

][m1r21 +m2 (r21 + r22 + 2r1r2 cosϕ)]2

dϕ ∧ (r2dr1 − r1dr2).

The local representation of the connection can be computed from the expression

α(θ1, r1, r2, ϕ)(θ1, r1, r2, ϕ)

= A(r1, r2, ϕ)

r1

r2

ϕ

+ θ1

=m2

m1r21 +m2(r21 + r22 + 2r1r2 cosϕ)

[−r2 sinϕ r1 sinϕ r22 + r1r2 cosϕ

]r1

r2

ϕ

+ θ1.

From this, we observe that

A(r1, r2, ϕ) =m2

m1r21 +m2(r21 + r22 + 2r1r2 cosϕ)

[−r2 sinϕ r1 sinϕ r22 + r1r2 cosϕ

].

The amended potential Vµ is given by

Vµ(q) = V (q) +12〈µ, I(q)−1µ〉

= [m1gq1 +m2g(q1 + q2)] · k +12· µ2

m1‖q⊥1 ‖2 +m2‖(q1 + q2)⊥‖2

= −m1g√l21 − r21 −m2g

(√l21 − r21 +

√l22 − r22

)+

12· µ2

m1r21 +m2(r21 + r22 + 2r1r2 cosϕ).

The Routhian has the following expression on the momentum level set,

Rµ =12‖hor(q, v)‖2 − Vµ.

Recall that hor(vq) = vq − ξQ(vq), where ξ = α(vq), and ξQ(vq) = (0, ξ, 0, 0, ξ, 0). Then, we obtain

hor(vq) = vq − (0, α(vq), 0, 0, α(vq), 0)

68

= (r1, θ1 − α(vq), z1, r2, θ2 − α(vq), z2).

From this, we conclude that

12‖hor(q, v)‖2

=12

r1

θ1 − α

z1

r2

θ2 − α

z2

T

m1 +m2 0 0 m2 cosϕ −m2r2 sinϕ 0

0 (m1 +m2)r21 0 m2r1 sinϕ m2r1r2 cosϕ 0

0 0 m1 +m2 0 0 0

m2 cosϕ m2r1 sinϕ 0 m2 0 0

−m2r2 sinϕ m2r1r2 cosϕ 0 0 m2r22 0

0 0 0 0 0 m2

r1

θ1 − α

z1

r2

θ2 − α

z2

=

12(m1 +m2)r21 + 2m2r1r2 cosϕ+m2r

22

α2

−m1r

21 θ1 +m2

[r1r2(θ1 + θ2) cosϕ+ (r1r2 − r2r1) sinϕ+ (r21 θ1 + r22 θ2)

]α

+12m1(r21 + r21 θ

21 + z2

1)

+12m2

r21 + r21 θ

21 + r22 + r22 θ

22 + 2(r1r2 + r1r2θ1θ2) cosϕ

+2(r1r2θ1 − r2r1θ2) sinϕ+ (z1 + z2)2,

where α = µI .

These combine to yield an expression for the Routhian Rµ, which drops to TS to give Rµ, and

allow us to apply the Reduced Symplectic Partitioned Runge–Kutta algorithm.

2.10.3 Example Trajectories

We have computed the reduced trajectory of the double spherical pendulum using the fourth-order

RSPRK algorithm on the Routh equations, and the fourth-order SPRK algorithm on the classical

Routh equations.

As discussed in §2.7, these two methods should yield equivalent reduced dynamics, related to

each other by a momentum shift, and in particular, their trajectories in position space should agree.

We first consider the evolution of r1, r2, and ϕ, using the RSPRK algorithm on the Routh equations,

as well as the projection of the relative position of m2 with respect to m1 onto the xy plane as seen

in Figure 2.7.

Figure 2.8 illustrates that the energy behavior of the trajectory is very good, as is typical of

variational integrators, and does not exhibit a spurious drift. In comparison, when a non-symplectic

fourth-order Runge–Kutta is applied to the unreduced dynamics, with time-steps that were a quarter

69

0 100 200 300 4000

100

200

300

400

500

600

700

t

r 1

0 100 200 300 4000

200

400

600

800

1000

1200

1400

t

r 20 100 200 300 400

0

5

10

15

t

φ

−1000 −500 0 500 1000 1500

−1000

−500

0

500

1000

x

y

Figure 2.7: Time evolution of r1, r2, ϕ, and the trajectory of m2, relative to m1, using RSPRK.

of that used in the symplectic method, and we notice a systematic drift in the energy behavior.

Finally, we consider the relative error between the position trajectories and energy obtained

from the RSPRK algorithm applied to the Routh equations as compared to the trajectories from

the SPRK algorithm applied to the classical Routh equations. As Figure 2.9 clearly illustrates, these

agree very well, as expected theoretically.

2.10.4 Computational Considerations

The choice of whether to compute in the unreduced space, and then project onto the shape space to

obtain the reduced dynamics, or to compute the reduced dynamics directly using either the Discrete

Routh equations, or the RSPRK algorithm, depends on the nature of the problem to be simulated.

Given a configuration space of dimension n, and a symmetry group of dimension m, we are faced

with the option of implementing a conceptually simpler algorithm in 2n dimensions, as compared

to a more geometrically involved algorithm in 2(n −m) dimensions. Whether the additional effort

associated with implementing the reduced algorithm is justified depend on a number of factors,

including the relative dimension of the configuration space and the symmetry group, the computa-

tional complexity of the iterative schemes used to solve the resulting implicit system of equations,

and any additional structure that may arise in either the reduced or unreduced system.

70

0 100 200 300 400

−8

−6

−4

−2

0

2x 10

−6

t

Rel

. Ene

rgy

Err

or

0 100 200 300 400

−8

−6

−4

−2

0

2x 10

−6

t

Rel

. Ene

rgy

Err

orFigure 2.8: Relative energy drift (E −E0)/E0 using RSPRK (left) compared to the relative energy

drift in a non-symplectic RK (right).

0 100 200 300 400−3

−2

−1

0

1

2x 10

−6

t

Rel

. Err

or in

r 1

0 100 200 300 400−5

−4

−3

−2

−1

0

1

2x 10

−6

t

Rel

. Err

or in

r 2

0 100 200 300 400−3

−2

−1

0

1

2

3x 10

−7

t

Rel

. Err

or in

φ

0 100 200 300 400

−10

−5

0

x 10−9

t

Rel

. Err

or in

E

Figure 2.9: Relative error in r1, r2, ϕ, and E between RSPRK and SPRK.

71

For instance, if longtime or repeated simulations are desired of systems with high-dimensional

symmetry groups, it can be advantageous to compute in the reduced space directly. An example

of this situation, which is of current engineering interest, is simulating the dynamical behavior of

connected networks of systems with their own internal symmetries.

If the systems to be connected are all identical, the geometric quantities that need to be computed,

such as the mechanical connection, have a particularly simple repeated form, and the additional

upfront effort in implementing the reduced algorithm can result in substantial computational savings.

Non-intrinsic numerical schemes such as the Symplectic Partitioned Runge–Kutta algorithm

applied to the classical Routh equations can have undesirable numerical properties due to the need

for coordinate-dependent local trivializations and the presence of coordinate singularities in these

local trivializations, such as those encountered while using Euler angles for rigid-body dynamics.

In the presence of non-trivial magnetic terms in the symplectic form, this can necessitate frequent

changes of coordinate charts, as documented in Wisdom et al. [1984] and Patrick [1991]. In such

instances, the coordinate changes can account for an overwhelming portion of the total computational

effort. In contrast, intrinsic methods do not depend on a particular choice of coordinate system, and

as such allow for the use of global charts through the use of containing vector spaces with constraints

enforced using Lagrange multipliers.

Coordinate singularities can affect the quality of the simulation in subtle ways that may depend

on the choice of numerical scheme. In the energy behavior of the simulation of the double spherical

pendulum, we notice spikes in the energy corresponding to times when r1 or r2 are close to 0. While

these errors accumulate in the non-symplectic method, the energy error in the symplectic method

remains well-behaved. However, sharp spikes can be avoided altogether by evolving the equations as

a constrained system with V = R3×R3, and constraint function g(v1,v2) = (‖v1‖−l1, ‖v2‖−l2) that

is imposed using Lagrange multipliers, as opposed to choosing local coordinates that automatically

satisfy the constraints. Here, the increased cost of working in the six-dimensional linear space V with

constraints is offset by not having to transform between charts of S2l1×S2

l2, which can be significant

if the trajectories are particularly chaotic.

While in simple examples, the effect of choosing local coordinates that allow the use of non-

intrinsic schemes can be properly corrected for, this is not true in general for more complicated

examples. Here, intrinsic schemes such as those we have developed in this paper for dealing with

reduced dynamics and constrained systems are preferable, since they do not depend on a particular

choice of local trivialization, and as such do not require frequent coordinate transformations.

72

2.11 Conclusions and Future Work

In summary, we have derived the Discrete Routh equations on S × S, which are symplectic with

respect to a non-canonical symplectic form, and retains the good energy behavior typically associated

with variational integrators. Furthermore, when the group action can be expressed as addition, we

obtain the Reduced Symplectic Partitioned Runge–Kutta algorithm on T ∗S, that can be considered

as a discrete analogue of cotangent bundle reduction. In addition, the theory has been extended to

include constraints and forcing. By providing an understanding of how the reduced and unreduced

formulations are related at a discrete level, we enable the user to freely choose whichever formulation

is most appropriate, and provides the most insight into the problem at hand.

Certainly one of the obvious things to do in the future is to extend this reduction procedure to

the case of nonabelian symmetry groups following the nonabelian version of Routh reduction given

in Jalnapurkar and Marsden [2000] and Marsden et al. [2000b]. There are also several problems,

including the averaged J2 problem, in which one can also carry out discrete reduction by stages and

in particular relate it to the semidirect product work of Bobenko and Suris [1999]. This is motivated

by the fact that the semidirect product reduction theory of Holm et al. [1998] is a special case

of reduction by stages (at least without the momentum map constraint), as was shown in Cendra

et al. [1998]. In further developing discrete reduction theory, the discrete theory of connections on

principal bundles developed in Leok et al. [2003] and Chapter 4 is particularly relevant, as it provides

an intrinsic method of representing the reduced space (Q×Q)/G as (S × S)⊕ G.

Another component that is needed in this work is a good discrete version of the calculus of

differential forms. Note that in our work we found, being directed by mechanics, that the right

discrete version of the magnetic 2-form is the difference of two connection 1-forms. It is expected

that we could recover such a magnetic 2-form by considering the discrete exterior derivative of a

discrete connection form in a finite discretization of space-time, and taking the continuum limit in

the spatial discretization. Developing a discrete analogue of Stokes’ Theorem would also provide

insight into the issue of discrete geometric phases. Some work on a discrete theory of exterior

calculus can be found in Desbrun et al. [2003a] and Chapter 3.

Of course, extensions of this work to the context of PDEs, especially fluid mechanics, would be

very interesting.

73

Chapter 3

Discrete Exterior Calculus

In collaboration with Mathieu Desbrun, Anil N. Hirani, and Jerrold E. Marsden.

Abstract

We present a theory and applications of discrete exterior calculus on simplicial com-

plexes of arbitrary finite dimension. This can be thought of as calculus on a discrete

space. Our theory includes not only discrete differential forms but also discrete vec-

tor fields and the operators acting on these objects. This allows us to address the

various interactions between forms and vector fields (such as Lie derivatives) which

are important in applications. Previous attempts at discrete exterior calculus have

addressed only differential forms. We also introduce the notion of a circumcentric

dual of a simplicial complex. The importance of dual complexes in this field has

been well understood, but previous researchers have used barycentric subdivision or

barycentric duals. We show that the use of circumcentric duals is crucial in arriving

at a theory of discrete exterior calculus that admits both vector fields and forms.

3.1 Introduction

This work presents a theory of discrete exterior calculus (DEC) motivated by potential appli-

cations in computational methods for field theories such as elasticity, fluids, and electromagnetism.

In addition, it provides much needed mathematical machinery to enable a systematic development

of numerical schemes that mirror the approach of geometric mechanics.

This theory has a long history that we shall outline below in §3.2, but we aim at a comprehensive,

systematic, as well as useful, treatment. Many previous works, as we shall review, are incomplete

both in terms of the objects that they treat as well as the types of meshes that they allow.

74

Our vision of this theory is that it should proceed ab initio as a discrete theory that parallels the

continuous one. General views of the subject area of DEC are common in the literature (see, for

instance, Mattiussi [2000]), but they usually stress the process of discretizing a continuous theory

and the overall approach is tied to this goal. However, if one takes the point of view that the discrete

theory can, and indeed should, stand in its own right, then the range of application areas naturally

is enriched and increases.

Convergence and consistency considerations alone are inadequate to discriminate between the

various choices of discretization available to the numerical analyst, and only by requiring, when

appropriate, that the discretization exhibits discrete analogues of continuous properties of interest

can we begin to address the question of what makes a discrete theory a canonical discretization of

a continuous one.

Applications to Variational Problems. One of the major application areas we envision is to

variational problems, be they in mechanics or optimal control. One of the key ingredients in this

direction that we imagine will play a key role in the future is that of AVI’s (asynchronous variational

integrators) designed for the numerical integration of mechanical systems, as in Lew et al. [2003].

These are integration algorithms that respect some of the key features of the continuous theory,

such as their multi-symplectic nature and exact conservation laws. They do so by discretizing the

underlying variational principles of mechanics rather than discretizing the equations. It is well-

known (see the reference just mentioned for some of the literature) that variational problems come

equipped with a rich exterior calculus structure and so on the discrete level, such structures will be

enhanced by the availability of a discrete exterior calculus. One of the objectives of this chapter is

to fill this gap.

Structured Constraints. There are many constraints in numerical algorithms that naturally

involve differential forms, such as the divergence constraint for incompressibility of fluids, as well as

the fact that differential forms are naturally the fields in electromagnetism, and some of Maxwell’s

equations are expressed in terms of the divergence and curl operations on these fields. Preserving,

as in the mimetic differencing literature, such features directly on the discrete level is another one

of the goals, overlapping with our goals for variational problems.

Lattice Theories. Periodic crystalline lattices are of important practical interest in material sci-

ence, and the anisotropic nature of the material properties arises from the geometry and connectivity

of the intermolecular bonds in the lattice. It is natural to model these lattices as inherently discrete

objects, and an understanding of discrete curvature that arises from DEC is particularly relevant,

since part of the potential energy arises from stretched bonds that can be associated with discrete

75

curvature in the underlying relaxed configuration of the lattice. In particular, this could yield a more

detailed geometric understanding of what happens at grain boundaries. Lattice defects can also be

associated with discrete curvature when appropriately interpreted. The introduction of a discrete

notion of curvature will lay the foundations for a better understanding of the role of geometry in

the material properties of solids.

Some of the Key Theoretical Accomplishments. Our development of discrete exterior cal-

culus includes discrete differential forms, the Hodge star operator, the wedge product, the exterior

derivative, as well as contraction and the Lie derivative. For example, this approach leads to the

proper definition of discrete divergence and curl operators and has already resulted in applications

like a discrete Hodge type decomposition of 3D vector fields on irregular grids—see Tong et al.

[2003].

Context. We present the theory and some applications of DEC in the context of simplicial com-

plexes of arbitrary finite dimension.

Methodology. We believe that the correct way to proceed with this program is to develop, as

we have already stressed, ab initio, a calculus on discrete manifolds which parallels the calculus on

smooth manifolds of arbitrary finite dimension. Chapters 6 and 7 of Abraham et al. [1988] are a

good source for the concepts and definitions in the smooth case. However we have tried to make

this chapter as self-contained as possible. Indeed, one advantage of developing a calculus on discrete

manifolds, as we do here, is pedagogical. By using concrete examples of discrete two- and three-

dimensional spaces one can explain most of calculus on manifolds at least formally as we will do

using the examples in this chapter. The machinery of Riemannian manifolds and general manifold

theory from the smooth case is, strictly speaking, not required in the discrete world. The technical

terms that are used in this introduction will be defined in subsequent sections, but they should be

already familiar to someone who knows the usual exterior calculus on smooth manifolds.

The Objects in DEC. To develop a discrete theory, one must define discrete differential forms

along with vector fields and operators involving these. Once discrete forms and vector fields are

defined, a calculus can be developed by defining the discrete exterior derivative (d), codifferential

(δ) and Hodge star (∗) for operating on forms, discrete wedge product (∧) for combining forms,

discrete flat ([) and sharp (]) operators for going between vector fields and 1-forms and discrete

contraction operator (iX) for combining forms and vector fields. Once these are done, one can then

define other useful operators. For example, a discrete Lie derivative (£X) can be defined by requiring

that the Cartan magic (or homotopy) formula hold. A discrete divergence in any dimension can

76

be defined. A discrete Laplace–deRham operator (∆) can be defined using the usual definition of

dδ + δd. When applied to functions, this is the same as the discrete Laplace–Beltrami operator

(∇2), which is the defined as div curl. We define all these operators in this chapter.

The discrete manifolds we work with are simplicial complexes. We will recall the standard formal

definitions in §3.3 but familiar examples of simplicial complexes are meshes of triangles embedded

in R3 and meshes made up of tetrahedra occupying a portion of R3. We will assume that the angles

and lengths on such discrete manifolds are computed in the embedding space RN using the standard

metric of that space. In other words, in this chapter we do not address the issue of how to discretize

a given smooth Riemannian manifold, and how to embed it in RN , since there may be many ways

to do this. For example, SO(3) can be embedded in R9 with a constraint, or as the unit quaternions

in R4. Another potentially important consideration in discretizing the manifold is that the topology

of the simplicial complex should be the same as the manifold to be discretized. This can be verified

using the methods of computational homology (see, for example, Kaczynski et al. [2004]), or discrete

Morse theory (see, for example, Forman [2002]; Wood [2003]). For the purposes of discrete exterior

calculus, only local metric information is required, and we will comment towards the end of §3.3

how to address the issue of embedding in a local fashion, as well as the criterion for a good global

embedding.

Our development in this chapter is for the most part formal in that we choose appropriate

geometric definitions of the various objects and quantities involved. For the most part, we do not

prove that these definitions converge to the smooth counterparts. The definitions are chosen so as to

make some important theorems like the generalized Stokes’ theorem true by definition. Moreover, in

the cases where previous results are available, we have checked that the operators we obtain match

the ones obtained by other means, such as variational derivations.

3.2 History and Previous Work

The use of simplicial chains and cochains as the basic building blocks for a discrete exterior cal-

culus has appeared in several papers. See, for instance, Sen et al. [2000], Adams [1996], Bossavit

[2002c], and references therein. These authors view forms as linearly interpolated versions of smooth

differential forms, a viewpoint originating from Whitney [1957], who introduced the Whitney and

deRham maps that establish an isomorphism between simplicial cochains and Lipschitz differential

forms.

We will, however, view discrete forms as real-valued linear functions on the space of chains.

These are inherently discrete objects that can be paired with chains of oriented simplices, or their

geometric duals, by the bilinear pairing of evaluation. In the next chapter, where we consider

77

applications involving the curvature of a discrete space, we will relax the condition that discrete

forms are real-valued, and consider group-valued forms.

Intuitively, this natural pairing of evaluation can be thought of as integration of the discrete form

over the chain. This difference from the work of Sen et al. [2000] and Adams [1996] is apparent in

the definitions of operations like the wedge product as well.

There is also much interest in a discrete exterior calculus in the computational electromagnetism

community, as represented by Bossavit [2001, 2002a,b,c], Gross and Kotiuga [2001], Hiptmair [1999,

2001a,b, 2002], Mattiussi [1997, 2000], Nicolaides and Wang [1998], Teixeira [2001], and Tonti [2002].

Many of the authors cited above, for example, Bossavit [2002c], Sen et al. [2000], and Hiptmair

[2002], also introduce the notions of dual complexes in order to construct the Hodge star operator.

With the exception of Hiptmair, they use barycentric duals. This works if one develops a theory of

discrete forms and does not introduce discrete vector fields. We show later that to introduce discrete

vector fields into the theory the notion of circumcentric duals seems to be important.

Other authors, such as Moritz [2000]; Moritz and Schwalm [2001]; Schwalm et al. [1999], have

incorporated vector fields into the cochain based approach to exterior calculus by identifying vector

fields with cochains, and having them supported on the same mesh. This is ultimately an unsatisfac-

tory approach, since dual meshes are essential as a means of encoding physically relevant phenomena

such as fluxes across boundaries.

The use of primal and dual meshes arises most often as staggered meshes in finite volume and

finite difference methods. In fluid computations, for example, the density is often a cell-centered

quantity, which can either be represented as a primal object by being associated with the 3-cell,

or as a dual object associated with the 0-cell at the center of the 3-cell. Similarly, the flux across

boundaries can be associated with the 2-cells that make up the boundary, or the 1-cell which is

normal to the boundary.

Another approach to a discrete exterior calculus is presented in Dezin [1995]. He defines a one-

dimensional discretization of the real line in much the same way we would. However, to generalize to

higher dimensions he introduces a tensor product of this space. This results in logically rectangular

meshes. Our calculus, however, is defined over simplicial meshes. A further difference is that like

other authors in this field, Dezin [1995] does not introduce vector fields into his theory.

A related effort for three-dimensional domains with logically rectangular meshes is that of Mans-

field and Hydon [2001], who established a variational complex for difference equations by constructing

a discrete homotopy operator. We construct an analogous homotopy operator for simplicial meshes

in proving the discrete Poincare lemma.

78

3.3 Primal Simplicial Complex and Dual Cell Complex

In constructing the discretization of a continuous problem in the context of our formulation of

discrete exterior calculus, we first discretize the manifold of interest as a simplicial complex. While

this is typically in the form of a simplicial complex that is embedded into Euclidean space, it is

only necessary to have an abstract simplicial complex, along with a local metric defined on adjacent

vertices. This abstract setting will be addressed further toward the end of this section.

We will now recall some basic definitions of simplices and simplicial complexes, which are standard

from simplicial algebraic topology. A more extensive treatment can be found in Munkres [1984].

Definition 3.1. A k-simplex is the convex span of k + 1 geometrically independent points,

σk = [v0, v1, . . . , vk] =

k∑i=0

αivi

∣∣∣∣∣αi ≥ 0,n∑i=0

αi = 1

.

The points v0, . . . , vk are called the vertices of the simplex, and the number k is called the dimen-

sion of the simplex. Any simplex spanned by a (proper) subset of v0, . . . , vk is called a (proper)

face of σk. If σl is a proper face of σk, we denote this by σl ≺ σk.

Example 3.1. Consider 3 non-collinear points v0, v1 and v2 in R3. Then, these three points indi-

vidually are examples of 0-simplices, to which an orientation is assigned through the choice of a sign.

Examples of 1-simplices are the oriented line segments [v0, v1], [v1, v2] and [v0, v2]. By writing the

vertices in that order we have given orientations to these 1-simplices, i.e., [v0, v1] is oriented from

v0 to v1. The triangle [v0, v1, v2] is a 2-simplex oriented in counterclockwise direction. Note that the

orientation of [v0, v2] does not agree with that of the triangle.

Definition 3.2. A simplicial complex K in RN is a collection of simplices in RN , such that,

1. Every face of a simplex of K is in K.

2. The intersection of any two simplices of K is a face of each of them.

Definition 3.3. A simplicial triangulation of a polytope |K| is a simplicial complex K such that

the union of the simplices of K recovers the polytope |K|.

Definition 3.4. If L is a subcollection of K that contains all faces of its elements, then L is a

simplicial complex in its own right, and it is called a subcomplex of K. One subcomplex of K is

the collection of all simplices of K of dimension at most k, which is called the k-skeleton of K,

and is denoted K(k).

79

Circumcentric Subdivision. We will also use the notion of a circumcentric dual or Voronoi

mesh of the given primal mesh. We will point to the importance of this choice later on in §3.7 and

3.9. We call the Voronoi dual a circumcentric dual since the dual of a simplex is its circumcenter

(equidistant from all vertices of the simplex).

Definition 3.5. The circumcenter of a k-simplex σk is given by the center of the k-circumsphere,

where the k-circumsphere is the unique k-sphere that has all k + 1 vertices of σk on its surface.

Equivalently, the circumcenter is the unique point in the k-dimensional affine space that contains the

k-simplex that is equidistant from all the k+1 nodes of the simplex. We will denote the circumcenter

of a simplex σk by c(σk).

The circumcenter of a simplex σk can be obtained by taking the intersection of the normals to

the (k−1)-dimensional faces of the simplex, where the normals are emanating from the circumcenter

of the face. This allows us to recursively compute the circumcenter.

If we are given the nodes which describe the primal mesh, we can construct a simplicial trian-

gulation by using the Delaunay triangulation, since this ensures that the circumcenter of a simplex

is always a point within the simplex. Otherwise we assume that a nice mesh has been given to us,

i.e., it is such that the circumcenters lie within the simplices. While this is not be essential for our

theory it makes some proofs simpler. For some computations the Delaunay triangulation is desirable

in that it reduces the maximum aspect ratio of the mesh, which is a factor in determining the rate

at which the corresponding numerical scheme converges. But in practice there are many problems

for which Delaunay triangulations are a bad idea. See, for example, Schewchuck [2002]. We will

address such computational issues in a separate work.

Definition 3.6. The circumcentric subdivision of a simplicial complex is given by the collection

of all simplices of the form

[c(σ0), . . . , c(σk)],

where σ0 ≺ σ1 ≺ . . . ≺ σk, or equivalently, that σi is a proper face of σj for all i < j.

Circumcentric Dual. We construct a circumcentric dual to a k-simplex using the circumcentric

duality operator, which is introduced below.

Definition 3.7. The circumcentric duality operator is given by

?(σk)

=∑

σk≺σk+1≺...≺σn

εσk,...,σn

[c(σk), c(σk+1), . . . , c(σn)

],

where the εσk,...,σn coefficient ensures that the orientation of[c(σk), c(σk+1), . . . , c(σn)

]is consistent

with the orientation of the primal simplex, and the ambient volume-form.

80

Orienting σk is equivalent to choosing a ordered basis, which we shall denote by dx1 ∧ . . . ∧

dxk. Similarly,[c(σk), c(σk+1), . . . , c(σn)

]has an orientation denoted by dxk+1 ∧ . . . ∧ dxn. If the

orientation corresponding to dx1∧ . . .∧dxn is consistent with the volume-form on the manifold, then

εσk,...,σn = 1, otherwise it takes the value −1.

We immediately see from the construction of the circumcentric duality operator that the dual

elements can be realized as a submesh of the first circumcentric subdivision, since it consists of

elements of the form [c(σ0), . . . , c(σk)], which are, by definition, part of the first circumcentric

subdivision.

Example 3.2. The circumcentric duality operator maps a 0-simplex into the convex hull generated

by the circumcenters of n-simplices that contain the 0-simplex,

?(σ0) =∑

ασnc (σn)∣∣∣ασn ≥ 0,

∑ασn = 1, σ0 ≺ σn

,

and the circumcentric duality operator maps a n−simplex into the circumcenter of the n−simplex,

?(σn) = c(σn).

This is more clearly illustrated in Figure 3.1, where the primal and dual elements are color coded

to represent the dual relationship between the elements in the primal and dual mesh.

(a) Primal (b) Dual (c) First subdivision

Figure 3.1: Primal, and dual meshes, as chains in the first circumcentric subdivision.

The choice of a circumcentric dual is significant, since it allows us to recover geometrically

important objects such as normals to (n− 1)-dimensional faces, which are obtained by taking their

circumcentric dual, whereas, if we were to use a barycentric dual, the dual to a (n− 1)-dimensional

face would not be normal to it.

81

Orientation of the Dual Cell. Notice that given an oriented simplex σk, which is represented

by [v0, . . . , vk], the orientation is equivalently represented by (v1− v0)∧ (v2− v1)∧ . . .∧ (vk − vk−1),

which we denote by,

[v0, . . . , vk] ∼ (v1 − v0) ∧ (v2 − v1) ∧ . . . ∧ (vk − vk−1),

which is an equivalence at the level of orientation. It would be nice to express our criterion for

determining the orientation of the dual cell in terms of the (k + 1)-vertex representation.

To determine the orientation of the (n−k)-simplex given by [c(σk), c(σk+1), . . . , c(σn)], or equiv-

alently, dxk+1∧ . . .∧dxn, we consider the n-simplex given by [c(σ0), . . . , c(σn)], where σ0 ≺ . . . ≺ σk.

This is related to the expression dx1 ∧ . . .∧ dxn, up to a sign determined by the relative orientation

of [c(σ0), . . . , c(σk)] and σk. Thus, we have that

dx1 ∧ . . . ∧ dxn ∼ sgn([c(σ0), . . . , c(σk)], σk)[c(σ0), . . . , c(σn)] .

Then, we need to check that dx1 ∧ . . . ∧ dxn is consistent with the volume-form on the manifold,

which is represented by the orientation of σn. Thus, we have that the correct orientation for the

[c(σk), c(σk+1), . . . , c(σn)] term is given by,

sgn([c(σ0), . . . , c(σk)], σk) · sgn([c(σ0), . . . , c(σn)], σn).

These two representations of the choice of orientation for the dual cells are equivalent, but the

combinatorial definition above might be preferable for the purposes of implementation.

Example 3.3. We would like to compute the orientation of the dual of a 1-simplex, in two dimen-

sions, given the orientation of the two neighboring 2-simplices.

Given a simplicial complex, as shown in Figure 3.2(a), we consider a 2-simplex of the form

[c(σ0), c(σ1), c(σ2)], which is illustrated in Figure 3.2(b).

Notice that the orientation is consistent with the given orientation of the 2-simplex, but it is not

consistent with the orientation of the primal 1-simplex, so the orientation should be reversed, to give

the dual cell illustrated in Figure 3.2(c).

We summarize the results for the induced orientation of dual cells for the other 2-simplices of

the form [c(σ0), c(σ1), c(σ2)], in Table 3.1.

Orientation of the Dual of a Dual Cell. While the circumcentric duality operator is a map from

the primal simplicial complex to the dual cell complex, we can formally extend the circumcentric

duality operator to a map from the dual cell complex to the primal simplicial complex. However,

82

(a) Simplicial complex (b) 2-simplex (c) ?σ1

Figure 3.2: Orienting the dual of a cell.

Table 3.1: Determining the induced orientation of a dual cell.

[c(σ0), c(σ1), c(σ2)]

sgn([c(σ0), c(σ1)], σ1) − + + −

sgn([c(σ0), c(σ1), c(σ2)], σ2) + − + −

sgn([c(σ0), c(σ1)], σ1)

· sgn([c(σ0), c(σ1), c(σ2)], σ2)

·[c(σ1), c(σ2)]

we need to be slightly careful about the orientation of primal simplex we recover from applying the

circumcentric duality operator twice.

We have that, ? ? (σk) = ±σk, where the sign is chosen to ensure the appropriate choice of

orientation. If, as before, σk has an orientation represented by dx1 ∧ . . . ∧ dxk, and ?σk has an

orientation represented by dxk+1 ∧ . . . ∧ dxn, then the orientation of ? ? (σk) is chosen so that

dxk+1∧ . . .∧dxn∧dx1∧ . . .∧dxk is consistent with the ambient volume-form. Since, by construction,

?(σk), dx1 ∧ . . . ∧ dxn has an orientation consistent with the ambient volume-form, we need only

compare dxk+1∧. . .∧dxn∧dx1∧. . .∧dxk with dx1∧. . .∧dxn. Notice that it takes n−k transpositions

to get the dx1 term in front of the dxk+1 ∧ . . .∧ dxn terms, and we need to do this k times for each

83

term of dx1 ∧ . . . ∧ dxk, so it follows that the sign is simply given by (−1)k(n−k), or equivalently,

? ? (σk) = (−1)k(n−k)σk. (3.3.1)

A similar relationship holds if we use a dual cell instead of the primal simplex σk.

Support Volume of a Primal Simplex and Its Dual Cell. We can think of a cochain as being

constructed out of a basis consisting of cosimplices or cocells with value 1 on a single simplex or cell,

and 0 otherwise. The way to visualize this cosimplex is that it is associated with a differential form

that has support on what we will refer to as the support volume associated with a given simplex

or cell.

Definition 3.8. The support volume of a simplex σk is a n-volume given by the convex hull of

the geometric union of the simplex and its circumcentric dual. This is given by

Vσk = convexhull(σk, ?σk) ∩ |K|.

The intersection with |K| is necessary to ensure that the support volume does not extend beyond

the polytope |K| which would otherwise occur if |K| is nonconvex.

We extend the notion of a support volume to a dual cell ?σk by similarly defining

V?σk = convexhull(?σk, ? ? σk) ∩ |K| = Vσk .

To clarify this definition, we will consider some examples of simplices, their dual cells, and their

corresponding support volumes. For two-dimensional simplicial complexes, this is illustrated in

Table 3.2.

The support volume has the nice property that at each dimension, it partitions the polytope

|K| into distinct non-intersecting regions associated with each individual k-simplex. For any two

distinct k-simplices, the intersection of their corresponding support volumes have measure zero, and

the union of the support volumes of all k-simplices recovers the original polytope |K|.

Notice, from our construction, that the support volume of a simplex and its dual cell are the

same, which suggests that there is an identification between cochains on k-simplices and cochains on

(n−k)-cells. This is indeed the case, and is a concept associated with the Hodge star for differential

forms.

Examples of simplices, their dual cells, and the corresponding support volumes in three dimen-

sions are given in Table 3.3.

In our subsequent discussion, we will assume that we are given a simplicial complex K of di-

84

Table 3.2: Primal simplices, dual cells, and support volumes in two dimensions.

Primal Simplex Dual Cell Support Volume

σ0, 0-simplex ?σ0, 2-cell Vσ0 = V?σ0



mension n in RN . Thus, the highest-dimensional simplex in the complex is of dimension n and

each 0-simplex (vertex) is in RN . One can obtain this, for example, by starting from 0-simplices,

i.e., vertices, and then constructing a Delaunay triangulation, using the vertices as sites. Often,

our examples will be for two-dimensional discrete surfaces in R3 made up of triangles (here n = 2

and N = 3) or three-dimensional manifolds made of tetrahedra, possibly embedded in a higher-

dimensional space.

Cell Complexes. The circumcentric dual of a primal simplicial complex is an example of a cell

complex. The definition of a cell complex follows.

Definition 3.9. A cell complex ?K in RN is a collection of cells in RN such that,

1. There is a partial ordering of cells in ?K, σk ≺ σl, which is read as σk is a face of σl.

2. The intersection of any two cells in ?K, is either a face of each of them, or it is empty.

3. The boundary of a cell is expressible as a sum of its proper faces.

We will see in the next section that the notion of boundary in the circumcentric dual has to be

modified slightly from the geometric notion of a boundary in order for the circumcentric dual to be

made into a cell complex.

85

Table 3.3: Primal simplices, dual cells, and support volumes in three dimensions.

Primal Simplex Dual Cell Support Volume





3.4 Local and Global Embeddings

While it is computationally more convenient to have a global embedding of the simplicial complex

into a higher-dimensional ambient space to account for non-flat manifolds it suffices to have an

abstract simplicial complex along with a local metric on vertices. The metric is local in the sense

that distances between two vertices are only defined if they are part of a common n-simplex in the

86

abstract simplicial complex. Then, the local metric is a map d : (v0, v1) | v0, v1 ∈ K(0), [v0, v1] ≺

σn ∈ K → R.

The axioms for a local metric are as follows,

Positive. d(v0, v1) ≥ 0, and d(v0, v0) = 0, ∀[v0, v1] ≺ σn ∈ K.

Strictly Positive. If d(v0, v1) = 0, then v0 = v1, ∀[v0, v1] ≺ σn ∈ K.

Symmetry. d(v0, v1) = d(v1, v0), ∀[v0, v1] ≺ σn ∈ K.

Triangle Inequality. d(v0, v2) ≤ d(v0, v1) + d(v1, v2), ∀[v0, v1, v2] ≺ σn ∈ K.

This allows us to embed each n-simplex locally into Rn, and thereby compute all the necessary metric

dependent quantities in our formulation. For example, the volume of a k-dual cell will be computed

as the sum of the k-volumes of the dual cell restricted to each n-simplex in its local embedding into

Rn.

This notion of local metrics and local embeddings is consistent with the point of view that exterior

calculus is a local theory with operators that operate on objects in the tangent and cotangent space

of a fixed point. The issue of comparing objects in different tangent spaces is addressed in the

discrete theory of connections on principal bundles in Leok et al. [2003] and Chapter 4.

This also provides us with a criterion for evaluating a global embedding. The embedding should

be such that the metric of the ambient space RN restricted to the vertices of the complex, thought of

as points in RN , agrees with the local metric imposed on the abstract simplicial complex. A global

embedding that satisfies this condition will produce the same numerical results in discrete exterior

calculus as that obtained using the local embedding method.

It is essential that the metric condition we impose is local, since the notion of distances between

points in a manifold which are far away is not a well-defined concept, nor is it particularly useful for

embeddings. As the simple example below illustrates, there may not exist any global embeddings

into Euclidean space that satisfies a metric constraint imposed for all possible pairs of vertices.

Example 3.4. Consider a circle, with the distance between two points given by the minimal arc

length. Consider a discretization given by 4 equidistant points on the circle, labelled v0, . . . , v3, with

the metric distances as follows,

d(vi, vi+1) = 1, d(vi, vi+2) = 2,

where the indices are evaluated modulo 4, and this distance function is extended to a metric on all

pairs of vertices by symmetry. It is easy to verify that this distance function is indeed a metric on

87

vertices.0 '!&"%#$

1 '!&"%#$

2 '!&"%#$????

????

? 3 '!&"%#$

?????????

1

2

If we only use the local metric constraint, then we only require that adjacent vertices are separated

by 1, and the following is an embedding of the simplicial complex into R2,

1

1

????

????

????

?

1

1

?????????????

If, however, we use the metric defined on all possible pairs of vertices, by considering v0, v1, v2, we

have that d(v0, v1) + d(v1, v2) = d(v0, v2). Since we are embedding these points into a Euclidean

space, it follows that v0, v1, v2 are collinear.

Similarly, by considering v0, v2, v3, we conclude that they are collinear as well, and that v1, v3

are coincident, which contradicts d(v1, v3) = 2. Thus, we find that there does not exist a global

embedding of the circle into Euclidean space if we require that the embedding is consistent with the

metric on vertices defined for all possible pairs of vertices.

3.5 Differential Forms and Exterior Derivative

We will now define discrete differential forms. We will use some terms (which we will define) from

algebraic topology, but it will become clear by looking at the examples that one can gain a clear

and working notion of what a discrete form is without any algebraic topology. We start with a few

definitions for which more details can be found on page 26 and 27 of Munkres [1984].

Definition 3.10. Let K be a simplicial complex. We denote the free abelian group generated by a

basis consisting of oriented k-simplices by Ck (K; Z) . This is the space of finite formal sums of the

k-simplices, with coefficients in Z. Elements of Ck(K; Z) are called k-chains.

Example 3.5. Figure 3.3 shows examples of 1-chains and 2-chains.

88

12 1

3

2−chain1−chain

•

•

•

••

•

•

•

•

•

•

????

??? yyyyyyyyyy

cccccccc

\\\\\\\\\\\ ****

****1

&&NNNNNNNN

3::vvvv 2

77

777

5

II

Figure 3.3: Examples of chains.

We view discrete k-forms as maps from the space of k-chains to R. Recalling that the space of

k-chains is a group, we require that these maps be homomorphisms into the additive group R. Thus,

discrete forms are what are called cochains in algebraic topology. We will define cochains below in

the definition of forms but for more context and more details readers can refer to any algebraic

topology text, for example, page 251 of Munkres [1984].

This point of view of forms as cochains is not new. The idea of defining forms as cochains

appears, for example, in the works of Adams [1996], Dezin [1995], Hiptmair [1999], and Sen et al.

[2000]. Our point of departure is that the other authors go on to develop a theory of discrete exterior

calculus of forms only by introducing interpolation of forms, which we will be able to avoid. The

formal definition of discrete forms follows.

Definition 3.11. A primal discrete k-form α is a homomorphism from the chain group Ck(K; Z)

to the additive group R. Thus, a discrete k-form is an element of Hom(Ck(K),R), the space of

cochains. This space becomes an abelian group if we add two homomorphisms by adding their

values in R. The standard notation for Hom(Ck(K),R) in algebraic topology is Ck(K; R). But we

will often use the notation Ωkd(K) for this space as a reminder that this is the space of discrete (hence

the d subscript) k-forms on the simplicial complex K. Thus,

Ωkd(K) := Ck(K; R) = Hom(Ck(K),R) .

Note that, by the above definition, given a k-chain∑i aic

ki (where ai ∈ Z) and a discrete k-form

α, we have that

α

(∑i

aicki

)=∑i

aiα(cki ) ,

and for two discrete k-forms α, β ∈ Ωkd(K) and a k-chain c ∈ Ck(K; Z),

(α+ β)(c) = α(c) + β(c) .

In the usual exterior calculus on smooth manifolds integration of k-forms on a k-dimensional

manifold is defined in terms of the familiar integration in Rk. This is done roughly speaking by

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doing the integration in local coordinates, and showing that the value is independent of the choice

of coordinates, due to the change of variables theorem in Rk. For details on this, see the first few

pages of Chapter 7 of Abraham et al. [1988]. We will not try to introduce the notion of integration

of discrete forms on a simplicial complex. Instead the fundamental quantity that we will work with

is the natural bilinear pairing of cochains and chains, defined by evaluation. More formally, we have

the following definition.

Definition 3.12. The natural pairing of a k-form α and a k-chain c is defined as the bilinear

pairing

〈α, c〉 = α(c).

As mentioned above, in discrete exterior calculus, this natural pairing plays the role that inte-

gration of forms on chains plays in the usual exterior calculus on smooth manifolds. The two are

related by a procedure done at the time of discretization. Indeed, consider a simplicial triangulation

K of a polyhedron in Rn, i.e., consider a “flat” discrete manifold. If we are discretizing a continuous

problem, we will have some smooth forms defined in the space |K| ⊂ Rn. Consider such a smooth

k-form αk. In order to define the discrete form αkd corresponding to αk, one would integrate αk on all

the k-simplices in K. Then, the evaluation of αkd on a k-simplex σk is defined by αkd(σk) :=

∫σk α

k.

Thus, discretization is the only place where integration plays a role in our discrete exterior calculus.

In the case of a non-flat manifold, the situation is somewhat complicated by the fact that the

smooth manifold, and the simplicial complex, as geometric sets embedded in the ambient space

do not coincide. A smooth differential form on the manifold can be discretized into the cochain

representation by identifying the vertices of the simplicial complex with points on the manifold, and

then using a local chart to identify k-simplices with k-volumes on the manifold.

There is the possibility of k-volumes overlapping even when their corresponding k-simplices do

not intersect, and this introduces a discretization error that scales like the mesh size. One can

alternatively construct geodesic boundary surfaces in an inductive fashion, which yields a partition

of the manifold, but this can be computationally prohibitive to compute.

Now we can define the discrete exterior derivative which we will call d, as in the usual exterior

calculus. The discrete exterior derivative will be defined as the dual, with respect to the natural

pairing defined above, of the boundary operator, which is defined below.

Definition 3.13. The boundary operator ∂k : Ck(K; Z) → Ck−1(K; Z) is a homomorphism defined

by its action on a simplex σk = [v0, . . . , vk],

∂kσk = ∂k([v0, . . . , vk]) =

k∑i=0

(−1)i[v0, . . . , vi, . . . , vk] ,

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where [v0, . . . , vi, . . . , vk] is the (k − 1)-simplex obtained by omitting the vertex vi. Note that ∂k

∂k+1 = 0.

Example 3.6. Given an oriented triangle [v0, v1, v2] the boundary, by the above definition, is the

chain [v1, v2]− [v0, v2] + [v0, v1], which are the three boundary edges of the triangle.

Definition 3.14. On a simplicial complex of dimension n, a chain complex is a collection of

chain groups and homomorphisms ∂k, such that,

0 // Cn(K)∂n // . . .

∂k+1// Ck(K)

∂k // . . . ∂1 // C0(K) // 0 ,

and ∂k ∂k+1 = 0.

Definition 3.15. The coboundary operator, δk : Ck(K) → CK+1(K), is defined by duality to

the boundary operator, with respect to the natural bilinear pairing between discrete forms and chains.

Specifically, for a discrete form αk ∈ Ωkd(K), and a chain ck+1 ∈ Ck+1(K; Z), we define δk by

⟨δkαk, ck+1

⟩=⟨αk, ∂k+1ck+1

⟩. (3.5.1)

That is to say

δk(αk) = αk ∂k+1 .

This definition of the coboundary operator induces the cochain complex,

0 Cn(K)oo . . .δn−1oo Ck(K)δk

oo . . .δk−1oo C0(K)δ0oo 0oo ,

where it is easy to see that δk+1 δk = 0.

Definition 3.16. The discrete exterior derivative denoted by d : Ωkd(K) → Ωk+1d (K) is defined

to be the coboundary operator δk.

Remark 3.1. With the above definition of the exterior derivative, d : Ωkd(K) → Ωk+1d (K), and the

relationship between the natural pairing and integration, one can regard equation 3.5.1 as a discrete

generalized Stokes’ theorem. Thus, given a k-chain c, and a discrete k-form α, the discrete

Stokes’ theorem, which is true by definition, states that

〈dα, c〉 = 〈α, ∂c〉 .

Furthermore, it also follows immediately that dk+1dk = 0.

91

Dual Discrete Forms. Everything we have said above in terms of simplices and the simplicial

complex K can be said in terms of the cells that are duals of simplices and elements of the dual

complex ?K. One just has to be a little more careful in the definition of the boundary operator, and

the definition we construct below is well-defined on the dual cell complex. This gives us the notion

of cochains of cells in the dual complex and these are the dual discrete forms.

Definition 3.17. The dual boundary operator, ∂k : Ck (?K; Z) → Ck−1 (?K; Z), is a homomor-

phism defined by its action on a dual cell σk = ?σn−k = ?[v0, . . . , vn−k],

∂σk = ∂ ? [v0, ..., vn−k]

=∑

σn−k+1σn−k

?σn−k+1 ,

where σn−k+1 is oriented so that it is consistent with the induced orientation on σn−k.

3.6 Hodge Star and Codifferential

In the exterior calculus for smooth manifolds, the Hodge star, denoted ∗, is an isomorphism between

the space of k-forms and (n−k)-forms. The Hodge star is useful in defining the adjoint of the exterior

derivative and this is adjoint is called the codifferential. The Hodge star, ∗ : Ωk(M) → Ωn−k(M), is

in the smooth case uniquely defined by the identity,

〈〈αk, βk〉〉v = αk ∧ ∗βk ,

where 〈〈 , 〉〉 is a metric on differential forms, and v is the volume-form. For a more in-depth discus-

sion, see, for example, page 411 of Abraham et al. [1988].

The appearance of k and (n− k) in the definition of Hodge star may be taken to be a hint that

primal and dual meshes will play some role in the definition of a discrete Hodge star, since the dual

of a k-simplex is an (n− k)-cell. Indeed, this is the case.

Definition 3.18. The discrete Hodge Star is a map ∗ : Ωkd(K) → Ωn−kd (?K), defined by its

action on simplices. For a k-simplex σk, and a discrete k-form αk,

1|σk|

〈αk, σk〉 =1

| ? σk|〈∗αk, ?σk〉.

The idea that the discrete Hodge star maps primal discrete forms to dual forms, and vice versa,

is well-known. See, for example, Sen et al. [2000]. However, notice we now make use of the volume

of these primal and dual meshes. But the definition we have given above does appear in the work

92

of Hiptmair [2002].

The definition implies that the primal and dual averages must be equal. This idea has already

been introduced, not in the context of exterior calculus, but in an attempt at defining discrete

differential geometry operators, see Meyer et al. [2002].

Remark 3.2. Although we have defined the discrete Hodge star above, we will show in Remark 3.8

of §3.12 that if an appropriate discrete wedge product and metric on discrete k-forms is defined, then

the expression for the discrete Hodge star operator follows from the smooth definition.

Lemma 3.1. For a k-form αk,

∗ ∗ αk = (−1)k(n−k)αk .

Proof. The proof is a simple calculation using the property that for a simplex or a cell σk, ?? (σk) =

(−1)k(n−k)σk (Equation 3.3.1).

Definition 3.19. Given a simplicial or a dual cell complex K the discrete codifferential oper-

ator, δ : Ωk+1d (K) → Ωkd(K), is defined by δ(Ω0

d(K)) = 0 and on discrete (k + 1)-forms to be

δβ = (−1)nk+1 ∗ d ∗ β .

With the discrete forms, Hodge star, d and δ defined so far, we already have enough to do an

interesting calculation involving the Laplace–Beltrami operator. But, we will show this calculation

in §3.9 after we have introduced discrete divergence operator.

3.7 Maps between 1-Forms and Vector Fields

Just as discrete forms come in two flavors, primal and dual (being linear functionals on primal chains

or chains made up of dual cells), discrete vector fields also come in two flavors. Before formally

defining primal and dual discrete vector fields, consider the examples illustrated in Figure 3.4. The

distinction lies in the choice of basepoints, be they primal or dual vertices, to which we assign vectors.

Definition 3.20. Let K be a flat simplicial complex, that is, the dimension of K is the same as

that of the embedding space. A primal discrete vector field X on a flat simplicial complex K

is a map from the zero-dimensional primal subcomplex K(0) (i.e., the primal vertices) to RN . We

will denote the space of such vector fields by Xd(K). The value of such a vector field is piecewise

constant on the dual n-cells of ?K. Thus, we could just as well have called such vector fields dual

and defined them as functions on the n-cells of ?K.

Definition 3.21. A dual discrete vector field X on a simplicial complex K is a map from the

zero-dimensional dual subcomplex (?K)(0) (i.e, the circumcenters of the primal n simplices) to RN

93

(a) Primal vector field (b) Dual vector field

Figure 3.4: Discrete vector fields.

such that its value on each dual vertex is tangential to the corresponding primal n-simplex. We

will denote the space of such vector fields by Xd(?K). The value of such a vector field is piecewise

constant on the n-simplices of K. Thus, we could just as well have called such vector fields primal

and defined them as functions on the n-simplices of K.

Remark 3.3. In this paper we have defined the primal vector fields only for flat meshes. We will

address the issue of non-flat meshes in separate work.

As in the smooth exterior calculus, we want to define the flat ([) and sharp (]) operators that

relate forms to vector fields. This allows one to write various vector calculus identities in terms of

exterior calculus.

Definition 3.22. Given a simplicial complex K of dimension n, the discrete flat operator on a

dual vector field, [ : Xd(?K) → Ωd(K), is defined by its evaluation on a primal 1 simplex σ1,

〈X[, σ1〉 =∑σnσ1

| ? σ1 ∩ σn|| ? σ1|

X · ~σ1 ,

where X · ~σ1 is the usual dot product of vectors in RN , and ~σ1 stands for the vector corresponding

to σ1, and with the same orientation. The sum is over all σn containing the edge σ1. The volume

factors are in dimension n.

Definition 3.23. Given a simplicial complex K of dimension n, the discrete sharp operator on

a primal 1-form, ] : Ωd(K) → Xd(?K), is defined by its evaluation on a given vertex v,

α](v) =∑

[v,σ0]

〈α, [v, σ0]〉∑

σn[v,σ0]

| ? v ∩ σn||σn|

n[v,σ0] ,

94

where the outer sum is over all 1-simplices containing the vertex v, and the inner sum is over all

n-simplices containing the 1-simplex [v, σ0]. The volume factors are in dimension n, and the vector

n[v,σ0] is the normal vector to the simplex [v, σ0], pointing into the n-simplex σn.

For a discussion of the proliferation of discrete sharp and flat operators that arise from considering

the interpolation of differential forms and vector fields, please see Hirani [2003].

3.8 Wedge Product

As in the smooth case, the wedge product we will construct is a way to build higher degree forms

from lower degree ones. For information about the smooth case, see the first few pages of Chapter

6 of Abraham et al. [1988].

Definition 3.24. Given a primal discrete k-form αk ∈ Ωkd(K), and a primal discrete l-form βl ∈

Ωld(K), the discrete primal-primal wedge product, ∧ : Ωkd(K)× Ωld(K) → Ωk+ld (K), defined by

the evaluation on a (k + l)-simplex σk+l = [v0, . . . , vk+l] is given by

〈αk ∧ βl, σk+l〉 =1

(k + l)!

∑τ∈Sk+l+1

sign(τ)|σk+l ∩ ?vτ(k)|

|σk+l|α ^ β(τ(σk+l)) ,

where Sk+l+1 is the permutation group, and its elements are thought of as permutations of the

numbers 0, . . . , k + l + 1. The notation τ(σk+l) stands for the simplex [vτ(0), . . . , vτ(k+l)]. Finally,

the notation α ^ β(τ(σk+l)) is borrowed from algebraic topology (see, for example, page 206 of

Hatcher [2001]) and is defined as

α ^ β(τ(σk+l)) := 〈α,[vτ(0), . . . , vτ(k)

]〉〈β,

[vτ(k), . . . , vτ(k+l)

]〉 .

Example 3.7. When we take the wedge product of two discrete 1-forms, we obtain terms in the

sum that are graphically represented in Figure 3.5.

Figure 3.5: Terms in the wedge product of two discrete 1-forms.

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Definition 3.25. Given a dual discrete k-form αk ∈ Ωkd(?K), and a primal discrete l-form βl ∈

Ωld(?K), the discrete dual-dual wedge product, ∧ : Ωkd(?K) × Ωld(?K) → Ωk+ld (?K), defined by

the evaluation on a (k + l)-cell σk+l = ?σn−k−l, is given by

〈αk ∧ βl, σk+l〉 =〈αk ∧ βl, ?σn−k−l〉

=∑

σnσn−k−l

sign(σn−k−l, [vk+l, . . . , vn])∑

τ∈Sk+l

sign(τ)

· 〈αk, ?[vτ(0), . . . , vτ(l−1), vk+l, . . . , vn]〉〈βl, ?[vτ(l), . . . , vτ(k+l−1), vk+l, . . . , vn]〉

where σn = [v0, . . . , vn], and, without loss of generality, assumed that σn−k−l = ±[vk+l, . . . , vn].

Anti-Commutativity of the Wedge Product.

Lemma 3.2. The discrete wedge product, ∧ : Ck(K) × Cl(K) → Ck+l(K), is anti-commutative,

i.e.,

αk ∧ βk = (−1)klβl ∧ αk .

Proof. We first rewrite the expression for the discrete wedge product using the following computa-

tion,

∑τ∈Sk+l+1

sign(τ)|σk+l ∩ ?vτ(k)|〈αk, τ [v0, . . . , vk]〉βl, τ [vk, . . . , vk+l]〉

=∑

τ∈Sk+l+1

(−1)k−1 sign(τ)|σk+l ∩ ?vτ(k)| 〈αk, τ [v1, . . . , v0, vk]〉〈βl, τ [vk, . . . , vk+l]〉

=∑

τ∈Sk+l+1

(−1)k−1 sign(τ)|σk+l ∩ ?vτρ(0)|〈αk, τρ[v1, . . . , vk, v0]〉〈βl, τρ[v0, vk+1, . . . , vk+l]〉

=∑

τ∈Sk+l+1

(−1)k−1(−1)k sign(τ)|σk+l ∩ ?vτρ(0)|〈αk, τρ[v0, . . . , vk]〉〈βl, τρ[v0, vk+1, . . . , vk+l]〉

=∑

τρ∈Sk+l+1ρ

(−1)k−1(−1)k(−1) sign(τ ρ)|σk+l ∩ ?vτρ(0)|

· 〈αk, τρ[v0, . . . , vk]〉〈βl, τρ[v0, vk+1, . . . , vk+l]〉

=∑

τ∈Sk+l+1

sign(τ)|σk+l ∩ ?vτ(0)|〈αk, τ [v0, . . . , vk]〉〈βl, τ [v0, vk+1, . . . , vk+l]〉 .

Here, we used the elementary fact, from permutation group theory, that a k+1 cycle can be written

as the product of k transpositions, which accounts for the (−1)k factors. Also, ρ is a transposition

96

of 0 and k. Then, the discrete wedge product can be rewritten as

〈αk ∧ βl, σk+l〉 =1

(k + l)!

∑τ∈Sk+l+1

sign(τ)|σk+l ∩ ?vτ(0)|

|σk+l|

· 〈αk, [vτ(0), . . . , vτ(k)]〉〈βl, [vτ(0),τ(k+1), . . . , vτ(k+l)]〉.

For ease of notation, we denote [v0, . . . , vk] by σk, and [v0, vk+1, . . . , vk+l] by σl. Then, we have

〈αk ∧ βl, σk+l〉 =1

(k + l)!

∑τ∈Sk+l+1


|σk+l|〈αk, τ(σk)〉〈βl, τ(σl)〉.

Furthermore, we denote [v0, vl+1 . . . , vk+l] by σk, and [v0, v1, . . . , vl] by σl. Then,

〈βl ∧ αk, σk+l〉 =1

(k + l)!

∑τ∈Sk+l+1


|σk+l|〈αk, τ(σk)〉〈βl, τ(σl)〉.

Consider the permutation θ ∈ Sk+l+1, given by

θ =

0 1 . . . k k + 1 . . . k + l

0 l + 1 . . . k + l 1 . . . l

,

which has the property that

σk = θ(σk),

σl = θ(σl).

Then, we have

〈βl ∧ αk, σk+l〉 =1

(k + l)!

∑τ∈Sk+l+1


|σk+l|〈αk, τ(σk)〉〈βl, τ(σl)〉

=1

(k + l)!

∑τ∈Sk+l+1

sign(τ)|σk+l ∩ ?vτθ(0)|

|σk+l|〈αk, τ θ(σk)〉〈βlτ θ(σl)〉

=1

(k + l)!

∑τθ∈Sk+l+1θ

sign(τ θ) sign(θ)|σk+l ∩ ?vτθ(0)|

|σk+l|〈αk, τ θ(σk)〉〈βl, τ θ(σl)〉.

By making the substitution, τ = τ θ, and noting that Sk+l+1θ = Sk+l+1, we obtain

〈βl ∧ αk, σk+l〉 = sign(θ)1

(k + l)!

∑τ∈Sk+l+1


|σk+l|〈αk, τ(σk)〉〈βl, τ(σl)〉

= sign(θ)〈αk ∧ βl, σk+l〉 .

97

To obtain the desired result, we simply need to compute the sign of θ, which is given by

sign(θ) = (−1)kl.

This follows from the observation that in order to move each of the last l vertices of σk+l forward,

we require k transpositions with v1, . . . , vk. Therefore, we obtain

〈βl ∧ αk, σk+l〉 = sign(θ)〈αk ∧ βl, σk+l〉 = (−1)kl〈αk ∧ βl, σk+l〉,

and

αk ∧ βl = (−1)klβl ∧ αk.

Leibniz Rule for the Wedge Product.

Lemma 3.3. The discrete wedge product satisfies the Leibniz rule,

d(αk ∧ βl) = (dαk) ∧ βl + (−1)kαk ∧ (dβl).

Proof. The proof of the Leibniz rule for discrete wedge products is directly analogous to the proof

of the coboundary formula for the simplicial cup product on cochains, which can be found on page

206 of Hatcher [2001]. This is because the discrete exterior derivative is precisely the coboundary

operator, and the wedge product is constructed out of weighted sums of cup products.

The cup product satisfies the Leibniz rule for an given partial ordering of the vertices, and the

permutations in the signed sum in the discrete wedge product correspond to different choices of

partial ordering. We then obtain the Leibniz rule for the discrete wedge product by applying it

term-wise for each choice of permutation.

Consider

〈(dαk) ∧ βl, σk+l+1〉 =k+1∑i=0

(−1)i1

(k + l)!

∑τ∈Sk+l+1


|σk+l|

· 〈αk, [vτ(0), . . . , vi, . . . , vτ(k+1)]〉〈βl, [vτ(k+1), . . . , vτ(k+l+1)]〉,

and

(−1)k〈αk ∧ (dβl), σk+l+1〉 = (−1)kk+l+1∑i=k

(−1)i−k1

(k + l)!

∑τ∈Sk+l+1


|σk+l|

· 〈αk, [vτ(0), . . . , vτ(k)]〉〈βl, [vτ(k), . . . , vi, . . . , vτ(k+l+1)]〉.

98

The last set of terms, i = k + 1, of the first expression cancels the first set of terms, i = k, of the

second expression, and what remains is simply 〈αk ∧ βl, ∂σk+l+1〉. Therefore, we can conclude that

〈(dαk) ∧ βl, σk+l+1〉+ (−1)k〈αk ∧ (dβl), σk+l+1〉 = 〈αk ∧ βl, ∂σk+l+1〉 = 〈d(αk ∧ βl), σk+l+1〉,

or simply that the Leibniz rule for discrete differential forms holds,

d(αk ∧ βl) = (dαk) ∧ βl + (−1)kαk ∧ (dβl).

Associativity for the Wedge Product. The discrete wedge product which we have introduced

is not associative in general. This is a consequence of the fact that the stencil for the two possible

triple wedge products are not the same. In the expression for 〈αk ∧ (βl∧γm), σk+l+m〉, each term in

the double summation consists of a geometric factor multiplied by 〈αk, σk〉〈βl, σl〉〈γm, σm〉 for some

k, l,m simplices σk, σl, σm.

Since βl and γm are wedged together first, σl and σm will always share a common vertex, but σk

could have a vertex in common with only σl, or only σm, or both. We can represent this in a graph,

where the nodes denote the three simplices, which are connected by an edge if, and only if, they

share a common vertex. The graphical representation of the terms which arise in the two possible

triple wedge products are given in Figure 3.6.

α ∧ (β ∧ γ) (α ∧ β) ∧ γ

Figure 3.6: Stencils arising in the double summation for the two triple wedge products.

For the wedge product to be associative for all forms, the two stencils must agree. Since the

stencils for the two possible triple wedge products differ, the wedge product is not associative in

general. However, in the case of closed forms, we can rewrite the terms in the sum so that all the

discrete forms are evaluated on triples of simplices that share a common vertex. This is illustrated

graphically in Figure 3.7.

This result is proved rigorously in the follow lemma.

Lemma 3.4. The discrete wedge product is associative for closed forms. That is to say, for αk ∈

99

Figure 3.7: Associativity for closed forms.

Ck(K), βl ∈ Cl(K), γm ∈ Cm(K), such that dαk = 0, dβl = 0, dγm = 0, we have that

(αk ∧ βl) ∧ γm = αk ∧ (βl ∧ γm).

Proof.

〈(αk ∧ βl) ∧ γm, σk+l+m〉

=∑

τ∈Sk+l+m+1

sign(τ)〈αk ∧ βl, τ [v0, . . . , vk+l]〉〈γm, τ [vk+l, . . . , vk+l+m]〉

=∑

τ∈Sk+l+m+1

∑ρ∈Sk+l+1

sign(τ) sign(ρ)〈αk, ρτ [v0, . . . , vk]〉

· 〈βl, ρτ [vk, . . . , vk+l]〉〈γm, τ [vk+l, . . . , vk+l+m]〉

Here, either ρτ(k) = τ(k+ l), in which case all three permuted simplices share vτ(k+l) as a common

vertex, or we need to rewrite either 〈αk, ρτ [v0, . . . , vk]〉 or 〈βl, ρτ [vk, . . . , vk+l]〉, using the fact that

αk and βl are closed forms.

If vτ(k+l) /∈ ρτ [v0, . . . , vk], then we need to rewrite 〈αk, ρτ [v0, . . . , vk]〉 by considering the simplex

obtained by adding the vertex vτ(k+l) to ρτ [v0, . . . , vk], which is [vτ(k+l), vρτ(0), . . . , vρτ(k)]. Then,

since αk is closed, we have that

0 = 〈dαk, [vτ(k+l), vρτ(0), . . . , vρτ(k)]〉

= 〈αk, ∂[vτ(k+l), vρτ(0), . . . , vρτ(k)]〉

= 〈αk, [vρτ(0), . . . , vρτ(k)]〉 −k∑i=0

(−1)i〈αk, [vτ(k+l), vρτ(0), . . . , vρτ(i), . . . , vρτ(k)]〉

100

or equivalently,

〈αk, [vρτ(0), . . . , vρτ(k)]〉 =k∑i=0

(−1)i〈αk, [vτ(k+l), vρτ(0), . . . , vρτ(i), . . . , vρτ(k)]〉.

Notice that all the simplices in the sum, with the exception of the last one, will share two vertices,

vτ(k+l) and vρτ(k) with ρτ [vk, . . . , vk+l], and so their contribution in the triple wedge product will

vanish due to the anti-symmetrized sum.

Similarly, if vτ(k+l) /∈ ρτ [vk, . . . , vk+l], using the fact that βl is closed yields

〈αk, [vρτ(k), . . . , vρτ(k+l)]〉 =k+l∑i=k

(−1)(i−k)〈αk, [vτ(k+l), vρτ(k), . . . , vρτ(i), . . . , vρτ(k+l)]〉.

As before, all the simplices in the sum, with the exception of the last one, will share two vertices,

vτ(k+l) and vρτ(k) with ρτ [v0, . . . , vk], and so their contribution in the triple wedge product will

vanish due to the anti-symmetrized sum.

This allows us to rewrite the triple wedge product in the case of closed forms as

〈(αk ∧ βl) ∧ γm, σk+l+m〉 =k+l+m∑i=0

∑τ∈Sk+l+m

sign(ρiτ)〈αk, ρiτ [v0, . . . , vk]〉〈βl, ρiτ [v0, vk+1, . . . , vk+l]〉

· 〈γm, ρiτ [v0, vk+l+1, . . . , vk+l+m]〉 ,

where τ ∈ Sk+l+m is thought of as acting on the set 1, . . . , k + l +m, and ρi is a transposition of

0 and i. A similar argument allows us to write αk ∧ (βl ∧ γm) in the same form, and therefore, the

wedge product is associative for closed forms.

Remark 3.4. This lemma is significant, since if we think of a constant smooth differential form,

and discretize it to obtain a discrete differential form, this discrete form will be closed. As such,

this lemma states that in the infinitesimal limit, the discrete wedge product we have defined will be

associative.

In practice, if we have a mesh with characteristic length ∆x, then we will have that

1|σk+l+m|

〈αk ∧ (βl ∧ γm)− (αk ∧ βl) ∧ γm, σk+l+m〉 = O(∆x),

which is to say that the average of the associativity defect is of the order of the mesh size, and

therefore vanishes in the infinitesimal limit.

101

3.9 Divergence and Laplace–Beltrami

In this section, we will illustrate the application of some of the DEC operations we have previously

defined to the construction of new discrete operators such as the divergence and Laplace–Beltrami

operators.

Divergence. The divergence of a vector field is given in terms of the Lie derivative of the volume-

form, by the expression, (div(X)µ = £Xµ. Physically, this corresponds to the net flow per unit

volume of an infinitesimal volume about a point.

We will define the discrete divergence by using the formulas defining them in the smooth exterior

calculus. The divergence definition will be valid for arbitrary dimensions. The resulting expressions

involve operators that we have already defined and so we can actually perform some calculations to

express these quantities in terms of geometric quantities. We will show that the resulting expression

in terms of geometric quantities is the same as that derived by variational means in Tong et al.

[2003].

Definition 3.26. For a discrete dual vector field X the divergence div(X) is defined to be

div(X) = −δX[ .

Remark 3.5. The above definition is a theorem in smooth exterior calculus. See, for example, page

458 of Abraham et al. [1988].

As an example, we will now compute the divergence of a discrete dual vector field on a two-

dimensional simplicial complex K, as illustrated in Figure 3.8.

Figure 3.8: Divergence of a discrete dual vector field.

A similar derivation works in higher dimensions, where one needs to be mindful of the sign that

arises from applying the Hodge star twice, ∗ ∗αk = (−1)k(n−k)αk. Since div(X) = −δX[, it follows

102

that div(X) = ∗d ∗X[. Since this is a primal 0-form it can be evaluated on a 0-simplex σ0, and we

have that

〈div(x), σ0〉 = 〈∗d ∗X[, σ0〉 .

Using the definition of discrete Hodge star, and the discrete generalized Stokes’ theorem, we get

1|σ0|

〈div(X), σ0〉 =1

| ? σ0|〈∗ ∗ d ∗X[, ?σ0〉

=1

| ? σ0|〈d ∗X[, ?σ0〉

=1

| ? σ0|〈∗X[, ∂(?σ0)〉 .

The second equality is obtained by applying the definition of the Hodge star, and the last equality

is obtained by applying the discrete generalized Stokes’ theorem. But,

∂(?σ0) =∑σ1σ0

?σ1 ,

as given by the expression for the boundary of a dual cell in Equation 3.17. Thus,

1|σ0|

〈div(X), σ0〉 =1

| ? σ0|〈∗X[,

∑σ1σ0

?σ1〉

=1

| ? σ0|∑σ1σ0

〈∗X[, ?σ1〉

=1

| ? σ0|∑σ1σ0

| ? σ1||σ1|

〈X[, σ1〉

=1

| ? σ0|∑σ1σ0

| ? σ1||σ1|

∑σ2σ1

| ? σ1 ∩ σ2|| ? σ1|

X · ~σ1

=1

| ? σ0|∑σ1σ0

∑σ2σ1

| ? σ1 ∩ σ2||σ1|

X · ~σ1

=1

| ? σ0|∑σ1σ0

| ? σ1 ∩ σ2| (X · ~σ1

|σ1|) .

This expression has the nice property that the divergence theorem holds on any dual n-chain,

which, as a set, is a simply connected subset of |K|. Furthermore, the coefficients we computed for

the discrete divergence operator are the unique ones for which a discrete divergence theorem holds.

Laplace–Beltrami. The Laplace–Beltrami operator is the generalization of the Laplacian to

curved spaces. In the smooth case the Laplace–Beltrami operator on smooth functions is defined to

be ∇2 = div curl = δd. See, for example, page 459 of Abraham et al. [1988]. Thus, in the smooth

103

case, the Laplace–Beltrami on functions is a special case of the more general Laplace–deRham op-

erator, ∆ : Ωk(M) → Ωk(M), defined by ∆ = dδ + δd.

As an example, we compute ∆f on a primal vertex σ0, where f ∈ Ω0d(K), and K is a (not

necessarily flat) triangle mesh in R3, as illustrated in Figure 3.9.

σ0

Figure 3.9: Laplace–Beltrami of a discrete function.

This calculation is done below.

1|σ0|

〈∆f, σ0〉 = 〈δdf, σ0〉

= −〈∗d ∗ df, σ0〉

= − 1| ? σ0|

〈d ∗ df, ?σ0〉

= − 1| ? σ0|

〈∗df, ∂(?σ0)〉

= − 1| ? σ0|

〈∗df,∑σ1σ0

?σ1〉

= − 1| ? σ0|

∑σ1σ0

〈∗df, ?σ1〉

= − 1| ? σ0|

∑σ1σ0

| ? σ1||σ1|

〈df, σ1〉

= − 1| ? σ0|

∑σ1σ0

| ? σ1||σ1|

(f(v)− f(σ0)) ,

where ∂σ1 = v−σ0. But, the above is the same as the formula involving cotangents found by Meyer

et al. [2002] without using discrete exterior calculus.

Another interesting aspect, which will be discussed in §3.12, is that the characterization of

harmonic functions as those functions which vanish when the Laplace–Beltrami operator is applied

104

is equivalent to that obtained from a discrete variational principle using DEC as the means of

discretizing the Lagrangian.

3.10 Contraction and Lie Derivative

In this section we will discuss some more operators that involve vector fields, namely contraction,

and Lie derivatives.

For contraction, we will first define the usual smooth contraction algebraically, by relating it to

Hodge star and wedge products. This yields one potential approach to defining discrete contraction.

However, since in the discrete theory we are only concerned with integrals of forms, we can use the

interesting notion of extrusion of a manifold by the flow of a vector field to define the integral of a

contracted discrete differential form.

We learned about this definition of contraction via extrusion from Bossavit [2002b], who goes

on to define discrete extrusion in his paper. Thus, he is able to obtain a definition of discrete

contraction. Extrusion turns out to be a very nice way to define integrals of operators involving

vector fields, and we will show how to define integrals of Lie derivatives via extrusion, which will

yield discrete Lie derivatives.

Definition 3.27. Given a manifold M , and S, a k-dimensional submanifold of M , and a vector

field X ∈ X(M), we call the manifold obtained by sweeping S along the flow of X for time t as the

extrusion of S by X for time t, and denote it by EtX(S). The manifold S carried by the flow for

time t will be denoted ϕtX(S).

Example 3.8. Figure 3.10 illustrates the 2-simplex that arises from the extrusion of a 1-simplex by

a discrete vector field that is interpolated using a linear shape function.

Figure 3.10: Extrusion of 1-simplex by a discrete vector field.

105

Contraction (Extrusion). We first establish an integral property of the contraction operator.

Lemma 3.5. ∫S

iXβ =d

dt

∣∣∣∣t=0

∫Et

X(S)

β

Proof. Prove instead that ∫ t

0

[∫Sτ

iXβ]dτ =

∫Et

X(S)

β .

Then, by first fundamental theorem of calculus, the desired result will follow. To prove the above,

simply take coordinates on S and carry them along with the flow and define the transversal coordinate

to be the flow of X. This proof is sketched in Bossavit [2002b].

This lemma allows us to interpret contraction as being the dual, under the integration pairing

between k-forms and k-volumes, to the geometric operation of extrusion. The discrete contraction

operator is then given by

〈iXαk+1, σk〉 =d

dt

∣∣∣∣t=0

〈αk+1, EtX(σk)〉,

where the evaluation of the RHS will typically require that the discrete differential form and the

discrete vector field are appropriately interpolated.

Remark 3.6. Since the dynamic definition of the contraction operator only depends on the derivative

of pairing of the differential form with the extruded region, it will only depend on the vector field in

the region S, and not on its extension into the rest of the domain.

In addition, if the interpolation for the discrete vector field satisfies a superposition principle,

then the discrete contraction operator will satisfy a corresponding superposition principle.

Contraction (Algebraic). Contraction is an operator that allows one to combine vector fields

and forms. For a smooth manifold M , the contraction of a vector field X ∈ X(M) with a (k+1)-form

α ∈ Ωk+1(M) is written as iXα, and for vector fields X1, . . . , Xk ∈ X(M), the contraction in smooth

exterior calculus is defined by

iXα(X1, . . . , Xk) = α(X,X1, . . . , Xk) .

We define contraction by using an identity that is true in smooth exterior calculus. This identity

originally appeared in Hirani [2003], and we state it here with proof.

Lemma 3.6 (Hirani [2003]). Given a smooth manifold M of dimension n, a vector field X ∈

X(M), and a k-form α ∈ Ωk(M), we have that

iXα = (−1)k(n−k) ∗ (∗α ∧X[) .

106

Proof. Recall that for a smooth function f ∈ Ω0(M), we have that iXα = f iXα. This, and the

multilinearity of α, implies that it is enough to show the result in terms of basis elements. In

particular, let τ ∈ Sn be a permutation of the numbers 1, . . . n, such that τ(1) < . . . < τ(k), and

τ(k + 1) < . . . < τ(n). Let X = eτ(j), for some j ∈ 1, . . . , n. Then, we have to show that

ieτ(j)eτ(1) ∧ . . . ∧ eτ(k) = (−1)k(n−k) ∗ (∗(eτ(1) ∧ . . . ∧ eτ(k)) ∧ eτ(j)) .

It is easy to see that the LHS is 0 if j > k, and it is

(−1)j−1(eτ(1) ∧ . . . ∧ eτ(j) . . . ∧ eσ(k)) ,

otherwise, where eτ(j) means that eτ(j) is omitted from the wedge product. Now, on the RHS of

Equation 3.6, we have that

∗(eτ(1) ∧ . . . ∧ eτ(k)) = sign(τ)(eτ(k+1) ∧ . . . ∧ eτ(n)) .

Thus, the RHS is equal to

(−1)k(n−k) sign(τ) ∗ (eτ(k+1) ∧ . . . ∧ eτ(n) ∧ eτ(j)) ,

which is 0 as required if j > k. So, assume that 1 ≤ j ≤ k. We need to compute

∗(eτ(k+1) ∧ . . . ∧ eτ(n) ∧ eτ(j)) ,

which is given by

s eτ(1) ∧ . . . eτ(j) ∧ . . . ∧ eτ(k) ,

where the sign s = ±1, such that the equation,

s eτ(k+1) ∧ . . . ∧ eτ(n) ∧ eτ(j) ∧ eτ(1) ∧ . . . ∧ eτ(j) ∧ . . . ∧ eτ(k) = µ ,

holds for the standard volume-form, µ = e1 ∧ . . . ∧ en. This implies that

s = (−1)j−1(−1)k(n−k) sign(τ) .

Then, RHS = LHS as required.

Since we have expressions for the discrete Hodge star (∗), wedge product (∧), and flat ([), we have

the necessary ingredients to use the algebraic expression proved in the above lemma to construct a

107

discrete contraction operator.

One has to note, however, that the wedge product is only associative for closed forms, and as a

consequence, the Leibniz rule for the resulting contraction operator will only hold for closed forms

as well. This is, however, sufficient to establish that the Leibniz rule for the discrete contraction will

hold in the limit as the mesh is refined.

Lie Derivative (Extrusion). As was the case with contraction, we will establish a integral

identity that allows the Lie derivative to be interpreted as the dual of a geometric operation on a

volume. This involves the flow of a volume by a vector field, and it is illustrated in the following

example.

Example 3.9. Figure 3.11 illustrates the flow of a 1-simplex by a discrete vector field interpolated

using a linear shape function.

Figure 3.11: Flow of a 1-simplex by a discrete vector field.

Lemma 3.7. ∫S

£Xβ =d

dt

∣∣∣∣t=0

∫ϕt

X(S)

β .

Proof.

F ∗t (£Xβ) =d

dtF ∗t β∫ t

0

F ∗τ (£Xβ)dτ = F ∗t β − β∫S

∫ t

0

F ∗τ (£Xβ)dτ =∫S

F ∗t β −∫S

β∫ t

0

∫ϕτ

X(S)

£Xβdτ =∫ϕt

X(S)

β −∫S

β .

108

This lemma allows us to define a discrete Lie derivative as follows,

〈£Xβk, σk〉 =

d

dt

∣∣∣∣t=0

〈βk, ϕtX(σk)〉 ,

where, as before, evaluating the RHS will require the discrete differential form and discrete vector

field to be appropriately interpolated.

Lie Derivative (Algebraic). Alternatively, as we have expressions for the discrete contraction

operator (iX), and exterior derivative (d), we can construct a discrete Lie derivative using the Cartan

magic formula,

£Xω = iXdω + diXω.

As is the case with the algebraic definition of the discrete contraction, the discrete Lie derivative

will only satisfy a Leibniz rule for closed forms. As before, this is sufficient to establish that the

Leibniz rule will hold in the limit as the mesh is refined.

3.11 Discrete Poincare Lemma

In this section, we will prove the discrete Poincare lemma by constructing a homotopy operator

though a generalized cocone construction. This section is based on the work in Desbrun et al.

[2003b].

The standard cocone construction fails at the discrete level, since the cone of a simplex is not,

in general, expressible as a chain in the simplicial complex. As such, the standard cocone does not

necessarily map k-cochains to (k − 1)-cochains.

An example of how the standard cone construction fails to map chains to chains is illustrated

in Figure 3.12. Given the simplicial complex on the left, consisting of triangles, edges and nodes,

we wish, in the center figure, to consider the cone of the bold edge with respect to the top most

node. Clearly, the resulting cone in the right figure, which is shaded grey, cannot be expressed as a

combination of the triangles in the original complex.

Figure 3.12: The cone of a simplex is, in general, not expressible as a chain.

109

In this subsection, a generalized cone operator that is valid for chains is developed which has the

essential homotopy properties to yield a discrete analogue of the Poincare lemma.

We will first consider the case of trivially star-shaped complexes, followed by logically star-shaped

complexes, before generalizing the result to contractible complexes.

Definition 3.28. Given a k-simplex σk = [v0, . . . , vk] we construct the cone with vertex w and base

σk, as follows,

w σk = [w, v0, . . . , vk].

Lemma 3.8. The geometric cone operator satisfies the following property,

∂(w σk) + w (∂σk) = σk.

Proof. This is a standard result from simplicial algebraic topology.

Trivially Star-Shaped Complexes.

Definition 3.29. A complex K is called trivially star-shaped if there exists a vertex w ∈ K(0),

such that for all σk ∈ K, the cone with vertex w and base σk is expressible as a chain in K. That

is to say,

∃w ∈ K(0) | ∀σk ∈ K,w σk ∈ Ck+1(K).

We can then denote the cone operation with respect to w as p : Ck(K) → Ck+1(K).

Lemma 3.9. In trivially star-shaped complexes, the cone operator, p : Ck(K) → Ck+1(K), satisfies

the following identity,

p∂ + ∂p = I,

at the level of chains.

Proof. Follows immediately from the identity for cones, and noting that the cone is well-defined at

the level of chains on trivially star-shaped complexes.

Definition 3.30. The cocone operator, H : Ck(K) → Ck−1(K), is defined by

〈Hαk, σk−1〉 = 〈αk, p(σk−1)〉.

This operator is well-defined on trivially star-shaped simplicial complexes.

Lemma 3.10. The cocone operator, H : Ck(K) → Ck−1(K), satisfies the following identity,

Hd + dH = I,

110

at the level of cochains.

Proof. A simple duality argument applied to the cone identity,

p∂ + ∂p = I,

yields the following,

〈αk, σk〉 = 〈αk, (p∂ + ∂p)σk〉

= 〈αk, p∂σk〉+ 〈αk, ∂pσk〉

= 〈Hαk, ∂σk〉+ 〈dαk, pσk〉

= 〈(dHαk, σk〉+ 〈Hdαk, σk〉

= 〈(dH +Hd)αk, σk〉.

Therefore,

Hd + dH = I,


Corollary 3.11 (Discrete Poincare Lemma for Trivially Star-shaped Complexes). Given

a closed cochain αk, that is to say, dαk = 0, there exists a cochain βk−1, such that, dβk−1 = αk.

Proof. Applying the identity for cochains,

Hd + dH = I,

we have,

〈αk, σk〉 = 〈(Hd + dH)αk, σk〉 ,

but, dαk = 0, so,

〈αk, σk〉 = 〈d(Hαk), σk〉.

Therefore, βk−1 = Hαk is such that dβk−1 = αk at the level of cochains.

Example 3.10. We demonstrate the construction of the tetrahedralization of the cone of a (n− 1)-

simplex over the origin.

111

If we denote by vki , the projection of the vi vertex to the k-th concentric sphere, where the 0-th

concentric sphere is simply the central point, then we fill up the cone [c, v1, ...vn] with simplices as

follows,

[v01 , v

11 , . . . , v

1n], [v

21 , v

11 , . . . , v

1n], [v

21 , v

22 , v

12 , . . . , v

1n], . . . , [v

21 , . . . , v

2n, v

1n].

Since Sn−1 is orientable, we can use a consistent triangulation of Sn−1 and these n-cones to con-

sistently triangulate Bn such that the resulting triangulation is star-shaped.

This fills up the region to the 1st concentric sphere, and we repeat the process by leapfrogging at

the last vertex to add [v21 , ..., v

2n, v

3n], and continuing the construction, to fill up the annulus between

the 1st and 2nd concentric sphere. Thus, we can keep adding concentric shells to create an arbitrarily

dense triangulation of a n-ball about the origin.

In three dimensions, these simplices are given by

[c, v11 , v

12 , v

13 ], [v2

1 , v11 , v

12 , v

13 ], [v2

1 , v22 , v

12 , v

13 ], [v2

1 , v22 , v

23 , v

13 ].

Putting them together, we obtain Figure 3.13.

Figure 3.13: Triangulation of a three-dimensional cone.

This example is significant, since we have demonstrated that for any n-dimensional ball about a

point, we can construct a trivially star-shaped triangulation of the ball, with arbitrarily high resolu-

tion. This allows us to recover the smooth Poincare lemma in the limit of an infinitely fine mesh,

using the discrete Poincare lemma for trivially star-shaped complexes.

112

Logically Star-Shaped Complexes.

Definition 3.31. A simplicial complex L is logically star-shaped if it is isomorphic, at the level

of an abstract simplicial complex, to a trivially star-shaped complex K.

Example 3.11. We see two simplicial complexes, in Figure 3.14, which are clearly isomorphic as

abstract simplicial complexes.

∼=

Figure 3.14: Trivially star-shaped complex (left); Logically star-shaped complex (right).

Definition 3.32. The logical cone operator p : Ck(L) → Ck+1(L) is defined by making the

following diagram commute,

Ck(K)pK // Ck+1(K)

Ck(L)pL // Ck+1(L)

Which is to say that, given the isomorphism ϕ : K → L, we define

pL = ϕ pK ϕ−1.

Example 3.12. We show an example of the construction of the logical cone operator.

pK //

pL //

113

This definition of the logical cone operator results in identities for the cone and cocone operator

that follow from the trivially star-shaped case, and we record the results as follows.

Lemma 3.12. In logically star-shaped complexes, the logical cone operator satisfies the following

identity,

p∂ + ∂p = I,


Proof. Follows immediately by pushing forward the result for trivially star-shaped complexes using

the isomorphism.

Lemma 3.13. In logically star-shaped complexes, the logical cocone operator satisfies the following

identity,

Hd + dH = I,


Proof. Follows immediately by pushing forward the result for trivially star-shaped complexes using

the isomorphism.

Similarly, we have a Discrete Poincare Lemma for logically star-shaped complexes.

Corollary 3.14 (Discrete Poincare Lemma for Logically Star-shaped Complexes). Given

a closed cochain αk, that is to say, dαk = 0, there exists a cochain βk−1, such that, dβk−1 = αk.

Proof. Follows from the above lemma using the proof for the trivially star-shaped case.

Contractible Complexes. For arbitrary contractible complexes, we construct a generalized cone

operator such that it satisfies the identity,

p∂ + ∂p = I,

which is the crucial property of the cone operator, from the point of view of proving the discrete

Poincare lemma.

The trivial cone construction gives a clue as to how to proceed in the construction of a gener-

alized cone operator. Notice that if a σk+1 is a term in p(σk), then p(σk+1) = ∅. This suggests

how we can use the cone identity to inductively construct the generalized cone operator.

To define p(σk), we consider σk+1 σk, such that, σk+1 and σk are consistently oriented. We

apply p∂ + ∂p to σk+1. Then, we have

σk+1 = p(σk) + p(∂σk+1 − σk) + ∂p(σk+1).

114

If we set p(σk+1) = ∅,

σk+1 = p(σk) + p(∂σk+1 − σk) + ∂(∅)

= p(σk) + p(∂σk+1 − σk).

Rearranging, we have

p(σk) = σk+1 − p(∂σk+1 − σk),

and

p(σk+1) = ∅.

We are done, so long as the simplices in the chain ∂σk+1 − σk already have p defined on it. This

then reduces to enumerating the simplices in such a way that in the right hand side of the equation,

we never evoke terms that are undefined.

We now introduce a method of augmenting a complex so that the enumeration condition is always

satisfied.

Definition 3.33. Given a n-complex K, consider a (n − 1)-chain cn−1 that is contained on the

boundary of K, and is included in the one-ring of some vertex on ∂K. Then, the one-ring cone

augmentation of K is the complex obtained by adding the n-cone w cn−1, and all its faces to the

complex.

Definition 3.34. A complex is generalized star-shaped if it can be constructed by repeatedly

applying the one-ring augmentation procedure.

We will explicitly show in Examples 3.13, and 3.16, how to enumerate the vertices in two and

three dimensions. And in Examples 3.15, and 3.17, we will introduce regular triangulations of R2

and R3 that can be constructed by inductive one-ring cone augmentation.

Remark 3.7. Notice that a non-contractible complex cannot be constructed by inductive one-ring

cone augmentation, since it will involve adding a cone to a vertex that has two disjoint base chains.

This prevents us from enumerating the simplices in such as way that all the terms in ∂σk+1 − σk

have had p defined on them, and we see in Example 3.18 how this causes the cone identity, and

hence the discrete Poincare lemma to break.

Example 3.13. In two dimensions, the one-ring condition implies that the base of the cone consists

of either one or two 1-simplices. To aid in visualization, consider Figure 3.15.

In the case of one 1-simplex, [v0, v1], when we augment using the cone construction with the new

115

Figure 3.15: One-ring cone augmentation of a complex in two dimensions.

vertex w, we define,

p([w]) = [v0, w] + p([v0]), p([v0, w]) = ∅,

p([v1, w]) = [v0, v1, w]− p([v0, v1]), p([v0, v1, w]) = ∅.

In the case of two 1-simplices, [v0, v1], [v0, v2], we have,

p([w]) = [v0, w] + p([v0]), p([v0, w]) = ∅,

p([v1, w]) = [v0, v1, w]− p([v0, v1]), p([v0, v1, w]) = ∅,

p([v2, w]) = [v0, v2, w]− p([v0, v2]), p([v0, v2, w]) = ∅.

Example 3.14. We will now explicitly utilize the one-ring cone augmentation procedure to compute

the generalized cone operator for part of a regular two-dimensional triangulation that is not logically

star-shaped.

As a preliminary, we shall consider a logically star-shaped complex, and augment with a new

vertex, as seen in Figure 3.16.

Figure 3.16: Logically star-shaped complex augmented by cone.

We use the logical cone operator for the subcomplex that is logically star-shaped, and the aug-

116

mentation rules in the example above for the newly introduced simplices. This yields,

p

= + p

= ,

p

= ∅,

p

= + p

= + ∅

= ,

p

= ∅,

p

= + p

= + = ,

p

= ∅.

Example 3.15. Clearly, the regular two-dimensional triangulation can be obtained by the successive

application of the one-ring cone augmentation procedure, as the following sequence illustrates,

117

7→ 7→ 7→ 7→ . . . ,

which means that the discrete Poincare lemma can be extended to the entire regular triangulation of

the plane.

Example 3.16. We consider the case of augmentation in three dimensions. Denote by v0, the center

of the one-ring on the two-surface, to which we are augmenting the new vertex w. The other vertices

of the one-ring are enumerated in order, v1, . . . , vm. To aid in visualization, consider Figure 3.17.

Figure 3.17: One-ring cone augmentation of a complex in three dimensions.

If the one-ring does not go completely around v0, we shall denote the missing term by [v0, v1, vm].

The generalized cone operators are given as follows.

k=0,

p([w]) = [v0, w] + p([v0]), p([v0, w]) = ∅,

k=1,

p([v1, w]) = [v0, v1, w]− p([v0, v1]), p([v0, v1, w]) = ∅,

p([vm, w]) = [v0, vm, w]− p([v0, vm]), p([v0, vm, w]) = ∅,

k=2,

p([v1, v2, w]) = [v0, v1, v2, w] + p([v0, v1, v2]), p([v1, v2, w]) = ∅,

118

p([vm−1, vm, w]) = [v0, vm−1, vm, w] + p([v0, vm−1, vm]), p([vm−1, vm, w]) = ∅.

If it does go around completely,

p([vm, v1, w]) = [v0, vm, v1, w] + p([v0, vm, v1]), p([v0, vm, v1, w]) = ∅.

Example 3.17. We provide a tetrahedralization of the unit cube that can be tiled to yield a regular

tetrahedralization of R3. The 3-simplices are as follows,

[v000, v001, v010, v10], [v001, v010, v100, v101], [v001, v010, v011, v101],

[v010, v100, v101, v110], [v010, v011, v101, v110], [v011, v101, v110, v111].

The tetrahedralization of the unit cube can be seen in Figure 3.18.

(a) Tileable tetrahedralization of the unit cube (b) Partial tiling of R3

Figure 3.18: Regular tiling of R3 that admits a generalized cone operator.

Since this regular tetrahedralization can be constructed by the successive application of the one-

ring cone augmentation procedure, the Discrete Poincare lemma can be extended to the entire regular

tetrahedralization of R3.

In higher dimensions, we can extend the construction of the generalized cone operator inductively

using the one-ring cone augmentation by choosing an appropriate enumeration of the base chain.

Topologically, the base chain will be the cone of Sn−2 (with possibly an open (n− 2)-ball removed)

119

with respect to the central point.

By spiraling around Sn−2, starting from around the boundary of the n−2 ball, and covering the

rest of Sn−2, as in Figure 3.19, we obtain the higher-dimensional generalization of the procedure we

have taken in Examples 3.13, and 3.16.

Figure 3.19: Spiral enumeration of Sn−2, n = 4.

Notice that n = 2 is distinguished, since S2−2 = S0 is disjoint, which is why in the two-

dimensional case, we were not able to use the spiraling technique to enumerate the simplices.

Since we have constructed the generalized cone operator such that the cone identity holds, we

have,

Lemma 3.15. In generalized star-shaped complexes, the generalized cone operator satisfies the fol-

lowing identity,

p∂ + ∂p = I,


Proof. By construction.

Lemma 3.16. In generalized star-shaped complexes, the generalized cocone operator satisfies the

following identity,

Hd + dH = I,


Proof. Follows immediately from applying the proof in the trivially star-shaped case, and using the

identity in the previous lemma.

Similarly, we have a discrete Poincare lemma for generalized star-shaped complexes.

120

Corollary 3.17 (Discrete Poincare Lemma for Generalized Star-shaped Complexes).

Given a closed cochain αk, that is to say, dαk = 0, there exists a cochain βk−1, such that, dβk−1 =

αk.

Proof. Follows from the above lemma using the proof for the trivially star-shaped case.

Example 3.18. We will consider an example of how the Poincare lemma fails in the case when the

complex is not contractible. Consider the following trivially star-shaped complex, and augment by

one vertex so as to make the region non-contractible, as show in Figure 3.20.

(a) Trivially star-shaped complex (b) Non-contractible complex

Figure 3.20: Counter-example for the discrete Poincare lemma for a non-contractible complex.

Now we attempt to verify the identity,

p∂ + ∂p = I,

and we will see how this is only true up to a chain that is homotopic to the inner boundary.

(p∂ + ∂p)

= p

+

-

+ ∂

= +

= +

121

Since the second term cannot be expressed as the boundary of a 2-chain, it will contribute a non-

trivial effect, even on closed discrete forms, and therefore the discrete Poincare lemma does not hold

for non-contractible complexes, as expected.

3.12 Discrete Variational Mechanics and DEC

We recall that discrete variational mechanics is based on a discrete analogue of Hamilton’s principle,

and they yield the discrete Euler–Lagrange equations. A particularly interesting property of DEC

arises when it is used to construct the discrete Lagrangian for harmonic functions, and Maxwell’s

equations.

In particular, for these examples, the following diagram commutes,

LagrangianL : TQ→ R

DEC //

Discrete LagrangianLd : Q×Q→ R

Euler–LagrangeEL : T 2Q→ T ∗Q

DEC //Discrete Euler–Lagrange

ELd : Q3 → T ∗Q

Which is to say that directly discretizing the differential equations for harmonic functions, and

Maxwell’s equations using DEC results in the same expressions as the discrete Euler–Lagrange equa-

tions associated with a discrete Lagrangian which is discretized from the corresponding continuous

Lagrangian by using DEC as the discretization scheme.

This is significant, since it implies that when DEC is used to discretize these equations, the

corresponding numerical scheme which is obtained is variational, and consequently exhibits excellent

structure-preserving properties.

In the variational principles for both harmonic functions and Maxwell’s equations, we require

the L2 norm obtained from the L2 inner product on Ωk(M), which is given by

〈αk, βk〉 =∫M

α ∧ ∗β .

The discrete analogue of this requires a primal-dual wedge product, which is given below for forms

of complementary dimension.

Definition 3.35. Given a primal discrete k-form αk ∈ Ωkd(K), and a dual discrete (n − k)-form

122

βn−k ∈ Ωn−kd (?K), the discrete primal-dual wedge product is defined as follows,

〈αk ∧ βn−k, Vσk〉 =|Vσk |

|σk|| ? σk|〈αk, σk〉〈βn−k, ?σk〉

=1n〈αk, σk〉〈βn−k, ?σk〉 ,

where Vσk is the n-dimensional support volume obtained by taking the convex hull of the simplex σk

and its dual cell ?σk.

The corresponding L2 inner product is as follows.

Definition 3.36. Given two primal discrete k-forms, αk, βk ∈ Ωkd(K), their discrete L2 inner

product, 〈αk, βk〉d, is given by

〈αk, βk〉d =∑σk∈K

|Vσk ||σk|| ? σk|

〈αk, σk〉〈∗β, ?σk〉

=1n

∑σk∈K

〈αk, σk〉〈∗β, ?σk〉 .

Remark 3.8. Notice that it would have been quite natural from the smooth theory to propose the

following metric tensor 〈〈 , 〉〉 for differential forms,

〈〈〈αk, βk〉〉v, Vσk〉 = |Vσk | 〈αk, σk〉|σk|

〈βk, σk〉|σk|

,

where the |Vσk | is the factor arising from integrating the volume-form over Vσk , and

〈αk, σk〉|σk|

〈βk, σk〉|σk|

is what we would expect for 〈〈αk, βk〉〉, if the forms αk and βk were constant on σk, which is the

product of the average values of αk, and βk.

If we adopt this as our definition of the metric tensor for forms, we can recover the definition we

obtained in §3.6 for the Hodge star operator. Starting from the definition from the smooth theory,

∫〈〈αk, βk〉〉v =

∫αk ∧ ∗βk ,

and expanding this in terms of the metric tensor for discrete forms, and the primal-dual wedge

operator, we obtain

〈〈〈αk, βk〉〉v, Vσk〉 = 〈αk ∧ ∗βk, Vσk〉 ,

123

|Vσk | 〈αk, σk〉|σk|

〈βk, σk〉|σk|

=|Vσk |

|σk|| ? σk|〈αk, σk〉〈∗βk, ?σk〉 .

When we eliminate common factors from both sides, we obtain the expression,

1|σk|

〈βk, σk〉 =1

| ? σk|〈∗βk, ?σk〉 ,

which is the expression we previously obtained in Definition 3.18 of §3.6.

The L2 norm for discrete differential forms is given below.

Definition 3.37. Given a primal discrete k-form αk ∈ Ωkd(K), its discrete L2 norm is given by

‖αk‖2d = 〈αk, αk〉d

=1n

∑σk∈K

〈αk, σk〉〈∗αk, ?σk〉

=1n

∑σk∈K

| ? σk||σk|

〈αk, σk〉2 .

Given these definitions, we can now reproduce some computations that were originally shown in

Castrillon-Lopez [2003].

Harmonic Functions. Harmonic functions φ : M → R can be characterized in a variational

fashion as extremals of the following action functional,

S(φ) =12

∫M

‖dφ‖2v,

where v is a Riemannian volume-form in M . The corresponding Euler–Lagrange equation is given

by

∗d ∗ dφ = −∆φ = 0,

which is the familiar characterization of harmonic functions in terms of the Laplace–Beltrami oper-

ator.

The discrete action functional can be expressed in terms of the L2 norm we introduced above

for discrete forms,

Sd(φ) =12‖dφ‖2d

=12n

∑σ1∈K

∣∣?σ1∣∣

|σ1|〈dφ, σ1〉2.

124

The basic variations needed for the determination of the discrete Euler–Lagrange operator are ob-

tained from variations that vary the value of the function φ at a given vertex v0, leaving the other

values fixed. These variations have the form,

φε = φ+ εη,

where η ∈ Ω0(M ; R) is such that 〈η, v0〉 = 1, and 〈η, v〉 = 0, for any v ∈ K(0) − v0. This family of

variations is enough to establish the variational principle. That is, we have

0 =d

dε

∣∣∣∣ε=0

Sd(φε)

=1n

∑σ1∈K

∣∣?σ1∣∣

|σ1|〈dφ, σ1〉〈dη, σ1〉

=1n

∑v0≺σ1

∣∣?σ1∣∣

|σ1|〈dφ, σ1〉 sgn(σ1; v0), (3.12.1)

where sgn(σ1; v) stands for the sign of σ1 with respect to v. Which is to say, sgn(σ1; v) = 1 if

σ1 = [v′, v], and sgn(σ1; v) = −1 if σ1 = [v, v′]. On the other hand,

〈∗d ∗ dφ, v0〉 =1

| ? v0|〈d ∗ dφ, ?v0〉

=1

| ? v0|〈∗dφ, ∂ ? v0〉

=1

| ? v0|∑v0≺σ1

〈∗dφ, ?σ1〉 sgn(σ1; v0)

=1

| ? v0|∑v0≺σ1

| ? σ1||σ1|

〈dφ, σ1〉 sgn(σ1; v0),

where in the second to last equality, one has to note that the border of the dual cell of a vertex v0

consists, up to orientation, in the dual of all the 1-simplices starting from v0. This is illustrated in

Figure 3.21, and follows from a general expression for the boundary of a dual cell that was given in

Definition 3.17.

The sign factor comes from the relation between the orientation of the dual of the 1-simplices

and that of ∂ ∗ v0. From this, we conclude that the variational discrete equation, given in Equation

3.12.1, is equivalent to the vanishing of the discrete Laplace–Beltrami operator,

∗d ∗ dφ = −∆ = 0.

Maxwell Equations. We can formulate the Maxwell equations of electromagnetism in a covariant

fashion by considering the 1-form A (the potential) as our fundamental variable in a Lorentzian

125

(a) Vertex v (b) ?v (c) ∂ ? v

(d) σ1 v (e) ?σ1

Figure 3.21: Boundary of a dual cell.

manifold X. The action functional for a Lagrangian formulation of electromagnetism is given by,

S(A) =12

∫X

‖dA‖2v ,

where ‖ · ‖ is the norm on forms induced by the Lorentzian metric on X, and v is the pseudo-

Riemannian volume-form. The 1-form A is related to the 4-vector potential encountered in the

relativistic formulation of electromagnetism (see, for example Jackson [1998]).

The Euler–Lagrange equation corresponding to this action functional is given by

∗d ∗ dA = 0 .

In terms of the field strength, F = dA, the last equation is usually rewritten as

dF = 0 , ∗d ∗ F = 0 ,

which is the geometric formulation of the Maxwell equations.

For the purposes of simplicity of exposition, we consider the special case where the Lorentzian

126

manifold decomposes into X = M × R, where (M, g) is a compact Riemannian 3-manifold. In

formulating the discrete version of this variational problem, we need to generalize the notion of a

discrete Hodge dual to take into account the pseudo-Riemannian metric structure. This can be

subtle in practice, and to overcome this, we consider a special family of complexes instead.

Let K ′ be a simplicial complex modelling M . For the sake of simplicity we consider M = R3

although this is not strictly necessary. We now consider a discretization tnn∈Z of R. We define

the complex K, modelling X = R4, the cells of which are the sets σ = σ′ × tn ⊂ R3 × R, and

σ = σ′ × (tn, tn+1) ⊂ R3 ×R for any σ′ ∈ K ′ and n ∈ Z. Of course, this is not a simplicial complex

but rather a “prismal” complex, as shown in Figure 3.22.

Space

Tim

e

Figure 3.22: Prismal cell complex decomposition of space-time.

The advantage of these cell complexes is the existence of the Voronoi dual. More precisely, given

any prismal cells σ′×tn ∈ K and σ′× (tn, tn+1) ∈ K, the Lorentz orthonormal to any of its edges

coincide with the Euclidean one in R4 and the existence of the circumcenter is thus guaranteed. In

other words, the Lorentz circumcentric dual ?K to K is the same as the Euclidean one in R4.

Remark 3.9. Much of the construction above can be carried out more generally by considering

arbitrary cell complexes in R4 that are not necessarily prismal, as long as none of its 1-cells are

lightlike. This causality condition is necessary to ensure that the circumcentric dual complex is well-

behaved. However, it is sufficient for computational purposes that the complex is well-centered, in

the sense that the Lorentzian circumcenter of each cell is contained inside the cell. These issues will

be addressed in future work.

Recall that the Hodge star ∗ is uniquely defined by satisfying the following expression,

α ∧ ∗β = 〈〈α, β〉〉v ,

for all α, β ∈ Ωk(X). The upshot of this is that the Hodge star operator depends on the metric, and

since we have a pseudo-Riemannian metric, there is a sign that is introduced in our expression for

127

the discrete Hodge star (Definition 3.18) that depends on whether the cell it is applied to is either

spacelike or timelike. The discrete Hodge star for prismal complexes in Lorentzian space is given

below.

Definition 3.38. The discrete Hodge star for prismal complexes in Lorentzian space is a

map ∗ : Ωkd(K) → Ωkd(∗K) defined by giving its action on cells in a prismal complex as follows,

1| ? σk|

〈∗αk, ?σk〉 = κ(σk)1|σk|

〈αk, σk〉,

where | · | stands for the volume and the causality sign κ(σk) is defined to be +1 if all the edges of

σk are spacelike, and −1 otherwise.

The causality sign of 2-cells in a (2 + 1)-space-time is summarized in Table 3.4. We should note

that the causality sign for a 0-simplex, κ(σ0), is always 1. This is because a 0-simplex has no edges,

and as such the statement that all of its edges are spacelike is trivially true.

Table 3.4: Causality sign of 2-cells in a (2 + 1)-space-time.

σ2

κ(σ2) +1 +1 −1 −1 −1

This causality term in the discrete Hodge star has consequences for the expression for the discrete

norm (Definition 3.37), which is now given.

Definition 3.39. Given a primal discrete k-form αk ∈ Ωkd(K), its discrete L2 Lorentzian norm

is given by,

‖αk‖2Lor,d =1n

∑σk∈K

〈αk, σk〉〈∗αk, ?σk〉

=1n

∑σk∈K

κ(σk)| ? σk||σk|

〈αk, σk〉2 .

Having defined the discrete Lorentzian norm, we can express the discrete action as

Sd(A) =12‖dA‖2Lor,d

128

=18

∑σ2∈K

〈dA, σ2〉〈∗dA, ?σ2〉

=18

∑σ2∈K

κ(σ2)| ? σ2||σ2|

〈dA, σ2〉2 .

The basic variations needed to determine the discrete Euler–Lagrange operator are obtained from

variations that vary the value of the 1-form A at a given 1-simplex σ10 , leaving the other values fixed.

These variations have the form,

Aε = Aε + εη,

where η ∈ Ω1d(K) is given by 〈η, σ1

0〉 = 1 for a fixed interior σ10 ∈ K and 〈η, σ1〉 = 0 for σ1 6= σ1

0 .

The derivation of the variational principle gives

d

dε

∣∣∣∣ε=0

Sd(Aε) =14

∑σ2∈K

| ? σ2||σ2|

κ(σ2)〈dA, σ2〉〈dη, σ2〉

=14

∑σ10≺σ2

| ? σ2||σ2|

κ(σ2)〈dA, σ2〉〈dη, σ2〉

=14

∑σ10≺σ2

| ? σ2||σ2|

κ(σ2)〈dA, σ2〉 sgn(σ2, σ10),

which vanishes for all the basic variations above. On the other hand, we now expand the discrete

1-form ∗d ∗ dA. For any σ10 ∈ K, we have that

〈∗d ∗ dA, σ10〉 =

|σ10 |

| ? σ10 |κ(σ1

0)〈d ∗ dA, ?σ10〉

=|σ1

0 || ? σ1

0 |κ(σ1

0)〈∗dA, ∂ ? σ10〉

=|σ1

0 || ? σ1

0 |κ(σ1

0)∑σ10≺σ2

sgn(σ2, σ10)〈∗dA, ?σ2〉

=|σ1

0 || ? σ1

0 |κ(σ1

0)∑σ10≺σ2

| ? σ2||σ2|

κ(σ2)〈dA, σ2〉 sgn(σ2, σ10),

where the sign sgn(σ2, σ1) stands for the relative orientation between σ2 and σ1. Which is to say,

sgn(σ2, σ1) = 1 if the orientation induced by σ2 on σ1 coincides with the orientation of σ1, and

sgn(σ2, σ1) = −1 otherwise. For the second to last equality, one has to note that the border of the

dual cell of an edge σ10 consists, conveniently oriented with the sgn operator, of the union of the

duals of all the 2-simplices containing σ10 . This statement is the content of Definition 3.17, which

gives the expression for the boundary of a dual cell, and was illustrated in Figure 3.21 for the case

of n-dimensional dual cells.

By comparing the two computations, we find that for an arbitrary choice of σ10 ∈ K, 〈∗d∗dA, σ1

0〉

129

is equal (up to a non-zero constant) to δSd(A), which always vanishes. It follows that the variational

discrete equations obtained above is equivalent to the discrete Maxwell equations,

∗d ∗ dA = 0.

3.13 Extensions to Dynamic Problems

It is desirable to leverage the exactness properties of the operators of discrete exterior calculus

to construct numerical algorithms with discrete conservation properties. For these purposes, it is

appropriate to extend the scope of DEC to incorporate dynamical behavior, by addressing the issue

of discrete diffeomorphisms and flows.

As discussed in the previous section, DEC and discrete mechanics have interesting synergis-

tic properties, and in this section we will explore a groupoid interpretation of discrete mechanics

that is particularly appropriate to formulating the notion of pull-back and push-forward of discrete

differential forms.

3.13.1 Groupoid Interpretation of Discrete Variational Mechanics

The groupoid formulation of discrete mechanics is particularly fruitful and natural, and it serves as

a unifying tool for understanding the variational formulation of discrete Lagrangian mechanics, and

discrete Euler–Poincare reduction, as discussed in the work of Weinstein [1996] and Marsden et al.

[1999, 2000a].

The groupoid interpretation of discrete mechanics is most clearly illustrated if we consider the

discretization of trajectories on TQ in two stages. Given a curve γ : R+ → TQ, we consider a

discrete sampling given by

gi = γ(ih) ∈ TQ.

We then approximate TQ by Q×Q, and associate to gi two elements in Q. We denote this by

gi 7→ (q0i , q1i ).

Or equivalently, in the language of groupoids, see Cannas da Silva and Weinstein [1999]; Weinstein

[2001], we have

G

α

β

Q

130

where α is the source map, and β is the target map. Then,

gi 7→ (α(gi), β(gi)) = (q0i , q1i ).

This can be visualized as

q0i = α(gi) q1i = β(gi)• •!!

gi

A product · : G(2) → G is defined on the set of composable pairs,

G(2) := (g, h) ∈ G×G | β(g) = α(h).

The groupoid composition g · h is defined by

α(g · h) = α(g),

β(g · h) = β(h).

This can be represented graphically as follows,

•α(g) = α(g · h)

•β(g) = α(h)

•β(h) = β(g · h)

g

!!

h

!!

g·h

!!

The set of composable pairs is the discrete analogue of the set of second-order curves on TQ. A

curve γ : R+ → TQ is said to be second-order if there exists a curve q : R+ → Q, such that,

γ(t) = (q(t), q(t)).

The corresponding condition for discrete curves is that given a sequence of points in Q × Q,

(q01 , q11), . . . , (q0p, q

1p), we require that

q1i = q0i+1.

This implies that the discrete curve on Q×Q is derived from a (p+ 1)-pointed curve (q0, . . . , qp) on

131

Q, where

qi =

q0i+1, if 0 ≤ i < p;

q1i , if i = p.

This condition has a direct equivalent in groupoids,

β(gi) = q1i = q0i+1 = α(gi+1).

Which is to say that the sequence of points in Q×Q are composable. In general, this hierarchy of

sets is denoted by

G(p) := (g1, . . . , gp) ∈ Gp | β (gi) = α (gi+1) ,

where G(0) ' Q.

In addition, the groupoid inverse is defined by the following,

α(g−1) = β(g),

β(g−1) = α(g).

This is represented as follows,

β(g−1) = α(g) α(g−1) = β(g)• •

g

!!

g−1

aa v

ng_XPI

Visualizing Groupoids. In summary, composition of groupoid elements, and the inverse of

groupoid elements can be illustrated by Figure 3.23. As we will see in the next subsection, rep-

resenting discrete diffeomorphisms as pair groupoids is the natural method of ensuring that the

mesh remains nondegenerate.

3.13.2 Discrete Diffeomorphisms and Discrete Flows

We will adopt the point of view of representing a discrete diffeomorphism as a groupoid, which was

first introduced in Pekarsky and West [2003], and appropriately modify it to reflect the simplicial

nature of our mesh. In addition, we will address the induced action of a discrete diffeomorphism on

the dual mesh.

Definition 3.40. Given a complex K embedded in V , and its corresponding abstract simplicial

complex M , a discrete diffeomorphism, ϕ ∈ Diffd(M), is a pair of simplicial complexes K1, K2,

132

β-fibersg

gh

h

β(g) = α(h)

α-fibers

g−1

G(0) ' Q

Figure 3.23: Groupoid composition and inverses.

which are realizations of M in the ambient space V . This is denoted by ϕ(M) = (K1,K2).

Definition 3.41. A one-parameter family of discrete diffeomorphisms is a map ϕ : I →

Diffd(M), such that,

π1(ϕ(t)) = π1(ϕ(s)), ∀s, t ∈ I.

Since we are concerned with evolving equations represented by these discrete diffeomorphisms,

and mesh degeneracy causes the numerics to fail, we introduce the notion of non-degenerate discrete

diffeomorphisms,

Definition 3.42. A non-degenerate discrete diffeomorphism ϕ = (K1,K2) is such that K1

and K2 are non-degenerate realizations of the abstract simplicial complex M in the ambient space

V .

Notice that it is sufficient to define the discrete diffeomorphism on the vertices of the abstract

complex M , since we can extend it to the entire complex by the relation

ϕ([v0, ..., vk]) = ([π1ϕ(v0), ..., π1ϕ(vk)], [π2ϕ(v0), ..., π2ϕ(vk)]).

If X ∈ K(0) is a material vertex of the manifold, corresponding to the abstract vertex w, that is

to say, π1ϕt(w) = X,∀t ∈ I, the corresponding trajectory followed by X in space is x = π2ϕt(w).

Then, the material velocity V (X, t) is given by

V (π1(w), t) =∂π2ϕs(w)

∂s

∣∣∣∣s=t

,

133

and the spatial velocity v(x, t) is given by

v(π2(w), t) = V (π1(w), t) =∂ϕs(ϕ−1

t (x))∂s

∣∣∣∣s=t

.

The distinction between the spatial and material representation is illustrated in Figure 3.24.

E1

E2

E3

X

e1

e2

e3

x

ϕ

Figure 3.24: Spatial and material representations.

The material velocity field can be thought of as a discrete vector field with the vectors based at

the vertices of K, which is to say that Tϕt ∈ Xd(K), is a discrete primal vector field. Notice that

ϕt on K induces a map ?ϕt on the vertices of the dual ?K, by the following,

?ϕt(c[v0, . . . , vn]) = (c[π1ϕt(v0), . . . , π1ϕt(vn)], c[π2ϕt(v0), . . . , π2ϕt(vn)]).

Similarly then, T ? ϕt ∈ Xd(?K) is a discrete dual vector field.

Comparison with Interpolatory Methods. At first glance, the groupoid formulation seems

like a cumbersome way to define a one-parameter family of discrete diffeomorphisms, and one may

be tempted to think of extending ϕt to the ambient space. We would then be thinking of ϕt : V → V .

This is undesirable since given ϕt and ψs which are non-degenerate flows, their composition ϕt ψs,

which is defined, may result in a degenerate mesh when applied to K. Thus, non-degenerate flows

are not closed under this notion of composition.

If we adopt groupoid composition instead at the level of vertices, we can always be sure that if

we compose two nondegenerate discrete diffeomorphisms, they will remain a nondegenerate discrete

diffeomorphism.

Discrete Diffeomorphisms as Pair Groupoids. The space of discrete diffeomorphisms natu-

rally has the structure of a pair groupoid. The discrete analogue of T Diff(M) from the point of

view of temporal discretization is the pair groupoid Diff(M) × Diff(M). In addition, we discretize

134

Diff(M) using Diffd(M), which is in turn a pair groupoid involving realizations of an abstract sim-

plicial complex in an ambient space.

3.13.3 Push-Forward and Pull-Back of Discrete Vector Fields and Dis-

crete Forms

For us to construct a discrete theory of exterior calculus that admits dynamic problems, it is critical

that we introduce the notion of push-forward and pull-back of discrete vector fields and discrete

forms under a discrete flow.

Push-Forward and Pull-Back of Discrete Vector Fields. The push-forward of a discrete

vector field satisfies the following commutative diagram,

K? //

f

?KX //

?f

RN

Tf

L? // ?L

f∗X // RN

and the pull-back satisfies the following commutative diagram,

K? //

f

?Kf∗X

//

?f

RN

Tf

L? // ?L

X // RN

By appropriately following the diagram around its boundary, we obtain the following expressions

for the push-forward and pull-back of a discrete vector field.

Definition 3.43. The push-forward of a dual discrete vector field X ∈ Xd(?K), under the

map f : K → L, is given by its evaluation on a dual vertex σ0 = ?σn ∈ (?L)(0),

f∗X(?σn) = Tf ·X(?(f−1(σn))).

Definition 3.44. The pull-back of a dual discrete vector field X ∈ Xd(?L), under the map

f : K → L, is given by its evaluation on a dual vertex σ0 = ?σn ∈ (?K)(0),

f∗X(?σn) = (f−1)∗X(?σn) = T (f−1) ·X(?(f(σn))).

Pull-Back and Push-Forward of Discrete Forms. A natural operation involving exterior

calculus in the context of dynamic problems is the pull-back of a differential form by a flow. We

135

define the pull-back of a discrete form as follows.

Definition 3.45. The pull-back of a discrete form αk ∈ Ωkd(L), under the map f : K → L, is

defined so that the change of variables formula holds,

〈f∗αk, σk〉 = 〈αk, f(σk)〉,

where σk ∈ K.

We can define the push-forward of a discrete form as its pull-back under the inverse map as

follows.

Definition 3.46. The push-forward of a discrete form αk ∈ Ωkd(K), under the map f : K → L

is defined by its action on σk ∈ L,

〈f∗αk, σk〉 = 〈(f−1)∗αk, σk〉 = 〈αk, f−1(σk)〉.

Naturality under Pull-Back of Wedge Product. We find that the discrete wedge product we

introduced in §3.8 is not natural under pull-back, which is to say that the relation

f∗(α ∧ β) = f∗α ∧ f∗β ,

does not hold in general. However, a metric independent definition that is natural under pull-back

was proposed in Castrillon-Lopez [2003].

Definition 3.47 (Castrillon-Lopez [2003]). Given a primal discrete k-form αk ∈ Ωkd(K), and

a primal discrete l-form βl ∈ Ωld(K), the natural discrete primal-primal wedge product,

∧ : Ωkd(K)×Ωld(K) → Ωk+ld (K), is defined by its evaluation on a (k+l)-simplex σk+l = [v0, . . . , vk+l],

〈αk ∧ βl, σk+l〉 =1

(k + l + 1)!

∑τ∈Sk+l+1

sign(τ)α ^ β(τ(σk+l)) .

In contrasting this definition to that given by Definition 3.24, we see that the geometric factor

|σk+l ∩ ?vτ(k)||σk+l|

,

has been replaced by1

k + l + 1

in this alternative definition. By replacing the geometric factor which is metric dependent with a

constant factor, Definition 3.47 becomes natural under pull-back.

136

The proofs in §3.8 that the discrete wedge product is anti-commutative, and satisfies a Leibniz

rule, remain valid for this alternative discrete wedge product, with only trivial modifications. As for

the proof of the associativity of the wedge product for closed forms, we note the following identity,

∑τ∈Sk+l+1

|σk+l ∩ ?vτ(k)||σk+l|

=∑

τ∈Sk+l+1

1k + l + 1

= (k + l)! ,

which is a crucial observation for the original proof to apply to the alternative wedge product.

3.14 Remeshing Cochains and Multigrid Extensions

It is sometimes desirable, particularly in the context of multigrid, multiscale, and multiresolution

computations, to be able to represent a discrete differential form which is given as a cochain on a

prescribed mesh, as one which is supported on a new mesh. Given a differential form ωk ∈ Ωk(K),

and a new mesh M such that |K| = |M |, we can define it at the level of cosimplices,

∀τk ∈M (k), 〈ωk, τk〉 =∑

σk∈K(k)

sgn(τk, σk)|Vτk ∩ Vσk ||Vσk |

〈ωk, σk〉,

and extend this by linearity to cochains. Here, sgn(τk, σk) is +1 if the orientation of τk and σk

are consistent, and −1 otherwise. Since k-skeletons of meshes that are not related by subdivision

may not have nontrivial intersections, intersections of support volumes are used in the remeshing

formula, as opposed to intersections of the k-simplices.

We denote this transformation at the level of cochains as, TK,M : Ck(K) → Ck(M). This has

the natural property that if we have a k-volume Uk that can be represented as a chain in either the

complex K or the complex M , that is to say, Uk = σk1 + . . .+ σkl = τk1 + . . .+ τkl , then we have

〈ωk, τk1 + . . .+ τkm〉 =m∑i=1

〈ω, τki 〉 =m∑i=1

∑σk∈K(k)

sgn(τki , σk)|Vτk

i∩ Vσk |

|Vσk |〈ωk, σk〉

=m∑i=1

l∑j=1

sgn(τki , σkj )|Vτk

i∩ Vσk

j|

|Vσkj|

〈ωk, σkj 〉

=l∑

j=1

m∑i=1

sgn(τki , σkj )|Vτk

i∩ Vσk

j|

|Vσkj|

〈ωk, σkj 〉

=l∑

j=1

〈ωk, σkj 〉 = 〈ωk, σk1 + . . .+ σkl 〉.

Which is to say that the integral of the differential form over Uk is well-defined, and independent of

the representation of the differential form.

137

Note that, in particular, if we choose to coarsen the mesh, the value the form takes on a cell in

the coarser mesh is simply the sum of the values the form takes on the old cells of the fine mesh

which make up the new cell in the coarser mesh.

Non-Flat Manifolds. The case of non-flat manifolds presents a challenge in remeshing akin to

that encountered in the discretization of differential forms. In particular, if the two meshes represent

different discretizations of a non-flat manifold, they will in general correspond to different polyhedral

regions in the embedding space, and not have the same support region.

We assume that our discretization of the manifold is sufficiently fine that for every simplex, all

its vertices are contained in some chart. Then, by using these local charts, we can identify support

volumes in the computational domain with n-volumes in the manifold, and thereby make sense of

the remeshing formula.


We have presented a framework for discrete exterior calculus using the cochain representation of

discrete differential forms, and introduced combinatorial representations of discrete analogues of

differential operators on discrete forms and discrete vector fields. The role of primal and dual cell

complexes in the theory are developed in detail. In addition, extensions to dynamic problems and

multi-resolution computations are discussed.

In the next few paragraphs, we will describe some of the future directions that emanate from the

current work on discrete exterior calculus.

Relation to Computational Algebraic Topology Since we have introduced a discrete Laplace-

deRham operator, one can hope to develop a discrete Hodge-deRham theory, and relate the deRham

cohomology of a simplicial complex to its simplicial cohomology.

Extensions to Non-Flat Manifolds. The intrinsic notion of what constitutes the discrete tan-

gent space to a node on a non-flat mesh remains an open question. It is possible that this notion

is related to a choice of discrete connection on the mesh, and it is an issue that deserves further

exploration.

Generalization to Arbitrary Tensors. The discretization of differential forms as cochains is

particularly natural, due to the pairing between forms and volumes by integration. When attempting

to discretize an arbitrary tensor, the natural discrete analogue is unclear. In particular, while it is

138

possible to expand an arbitrary tensor using the tensor product of covariant and contravariant one-

tensors, this would be cumbersome to represent on a mesh. In Chapter 4, which is on discrete

connections, we will see Lie group-valued discrete 1-forms, and one possible method of discretizing a

(p, q)-tensor that is alternating in the contravariant indices, is to consider it as a (0, q)-tensor-valued

discrete p-form.

It would be particularly interesting to explore this in the context of the elasticity complex (see,

for example, Arnold [2002]),

se(3) // C∞(Ω,R3) ε // C∞(Ω,S) J // C∞(Ω,S) div // C∞(Ω,R3) // 0 ,

where S is the space of 3 × 3 symmetric matrices. One approach to discretize this was suggested

in Arnold [2002], which cites the use of the Bernstein–Gelfand–Gelfand resolution in Eastwood [2000]

to derive the elasticity complex from the deRham complex. Alternatively, it might be appropriate

in the context of the elasticity complex to consider Lie algebra-valued discrete differential forms.

Convergence and Higher-Order Theories. The natural question from the point of view of nu-

merical analysis would be to carefully analyze the convergence properties of these discrete differential

geometric operators. In addition, higher-order analogues of the discrete theory of exterior calculus

are desirable from the point of view of computational efficiency, but the cochain representation is

attractive due to its conceptual simplicity and the elegance of representing discrete operators as

combinatorial operations on the mesh.

It would therefore be desirable to reconcile the two, by ensuring that high-order interpolation

and combinatorial operations are consistent. As a low-order example, Whitney forms, which are

used to interpolate differential forms on a simplicial mesh, have the nice property that taking the

Whitney form associated with the coboundary of a simplicial cochain is equal to taking the exterior

derivative of the Whitney form associated with the simplicial cochain. As such, the coboundary

operation, which is a combinatorial operation akin to finite differences, is an exact discretization of

the exterior derivative, when applied to the degrees of freedom associated to the finite-dimensional

function space of Whitney forms.

It would be interesting to apply subdivision surface techniques to construct interpolatory spaces

that are compatible with differential geometric operations that are combinatorial operations on the

degrees of freedom. This will result in a massively simplified approach to higher-order theories

of discrete exterior calculus, by avoiding the use of symbolic computation, which would otherwise

be necessary to compute the action of continuous exterior differential operators on the polynomial

expansions for differential forms.

139

Chapter 4

Discrete Connections on Principal Bundles

In collaboration with Jerrold E. Marsden, and Alan D. Weinstein.

Abstract

Connections on principal bundles play a fundamental role in expressing the equations

of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete

theory of connections on principal bundles is constructed by introducing the discrete

analogue of the Atiyah sequence, with a connection corresponding to the choice of a

splitting of the short exact sequence. Equivalent representations of a discrete connec-

tion are considered, and an extension of the pair groupoid composition, that takes

into account the principal bundle structure, is introduced. Computational issues,

such as the order of approximation, are also addressed. Discrete connections provide

an intrinsic method for introducing coordinates on the reduced space for discrete

mechanics, and provide the necessary discrete geometry to introduce more general

discrete symmetry reduction. In addition, discrete analogues of the Levi-Civita con-

nection, and its curvature, are introduced by using the machinery of discrete exterior

calculus, and discrete connections.

4.1 Introduction

One of the major goals of geometric mechanics is the study of symmetry, and its consequences. An

important tool in this regard is the non-singular reduction of mechanical systems under the action

of free and proper symmetries, which is naturally formulated in the setting of principal bundles.

The reduction procedure results in the decomposition of the equations of motion into terms

involving the shape and group variables, and the coupling between these are represented in terms

140

of a connection on the principal bundle.

Connections and their associated curvature play an important role in the phenomena of geometric

phases. A discussion of the history of geometric phases can be found in Berry [1990]. Shapere and

Wilczek [1989] is a collection of papers on the theory and application of geometric phases to physics.

In the rest of this section, we will survey some of the applications of geometric phases and connections

to geometric mechanics and control, some of which were drawn from Marsden [1994, 1997]; Marsden

and Ratiu [1999].

The simulation of these phenomena requires the construction of a discrete notion of connections

on principal bundles that is compatible with the approach of discrete variational mechanics, and it

towards this end that this chapter is dedicated.

Falling Cat. Geometric phases arise in nature, and perhaps the most striking example of this is

the falling cat, which is able to reorient itself by 180, while remaining at zero angular momentum,

as show in Figure 4.1.

The key to reconciling this with the constancy of the angular momentum is that angular momen-

tum depends on the moment of inertia, which in turn depends on the shape of the cat. When the

cat changes it shape by curling up and twisting, its moment of inertia changes, which is in turn com-

pensated by its overall orientation changing to maintain the zero angular momentum condition. The

zero angular momentum condition induces a connection on the principal bundle, and the curvature

of this connection is what allows the cat to reorient itself.

A similar experiment can be tried on Earth, as described on page 10 of Vedral [2003]. This

involves standing on a swivel chair, lifting your arms, and rotating them over your head, which will

result in the chair swivelling around slowly.

Holonomy. The sense in which curvature is related to geometric phases is most clearly illustrated

by considering the parallel transport of a vector around a curve on the sphere, as shown in Figure 4.2.

Think of the point on the sphere as representing the shape of the cat, and the vector as repre-

senting its orientation. The fact that the vector experiences a phase shift when parallel transported

around the sphere is an example of holonomy. In general, holonomy refers to a situation in ge-

ometry wherein an orthonormal frame that is parallel transported around a closed loop, back to its

original position, is rotated with respect to its original orientation.

Curvature of a space is critically related to the presence of holonomy. Indeed, curvature should

be thought of as being an infinitesimal version of holonomy, and this interpretation will resurface

when considering the discrete analogue of curvature in the context of a discrete exterior calculus.

141

c© Gerard Lacz/Animals Animals

Figure 4.1: Reorientation of a falling cat at zero angular momentum.

142

start

finisharea = A

Figure 4.2: A parallel transport of a vector around a spherical triangle produces a phase shift.

Foucault Pendulum. Another example relating geometric phases and holonomy is that of the

Foucault pendulum. As the Earth rotates about the Sun, the Foucault pendulum exhibits a phase

shift of ∆θ = 2π cosα (where α is the co-latitude). This phase shift is geometric in nature, and

is a consequence of holonomy. If one parallel transports an orthonormal frame around the line of

constant latitude, it exhibits a phase shift that is identical to that of the Foucault pendulum, as

illustrated in Figure 4.3.

parallel translate framealong a line of latitude

cut andunroll cone

Figure 4.3: Geometric phase of the Foucault pendulum.

True Polar Wander. A particular striking example of the consequences of geometric phases and

the conservation of angular momentum is the phenomena of true polar wander, that was studied by

143

Goldreich and Toomre [1969], and more recently by Leok [1998]. It is thought that some 500 to 600

million years ago, during the Vendian–Cambrian transition, the Earth, over a 15-million-year period,

experienced an inertial interchange true polar wander event. This occurred when the intermediate

and maximum moments of inertia crossed due to the redistribution of mass anomalies, associated

with continental drift and mantle convection, thereby causing a catastrophic shift in the axis of

rotation.

This phenomena is illustrated in Figure 4.4, wherein the places corresponding to the North and

South poles of the Earth migrate towards the equator as the axis of rotation changes.

Figure 4.4: True Polar Wander. Red axis corresponds to the original rotational axis, and the gold

axis corresponds to the instantaneous rotational axis.

Geometric Control Theory. Geometric phases also have interesting applications and conse-

quences in geometric control theory, and allow, for example, astronauts in free space to reorient

themselves by changing their shape. By holding one of their legs straight, swivelling at the hip, and

moving their foot in a circle, they are able to change their orientation. Since the reorientation only

occurs as the shape is being changed, this allows the reorientation to be done with extremely high

precision. Such ideas have been applied to the control of robots and spacecrafts; see, for example,

Walsh and Sastry [1993]. The role of connections in geometric control is also addressed in-depth

in Marsden [1994, 1997].

One of the theoretical underpinnings of the application of geometric phases to geometric control

was developed in Montgomery [1991] and Marsden et al. [1990], in the form of the rigid-body phase

formula,

∆θ =1‖µ‖

∫D

ωµ + 2HµT

= −Λ +

2HµT

‖µ‖,

144

the geometry of which is illustrated in Figure 4.5.

Pµ

geometric phase

dynamic phase

true trajectory

horizontal lift

reduced trajectory

πµ

Pµ

D

Figure 4.5: Geometry of rigid-body phase.

An example that has been studied extensively is that of the satellite with internal rotors, with a

configuration space given by Q = SE(3)× S1 × S1 × S1, and illustrated in Figure 4.6.

spinning rotors

rigid carrier

Figure 4.6: Rigid body with internal rotors.

The generalization of the rigid-body phase formula in the presence of feedback control is partic-

ularly useful in the study and design of attitude control algorithms.

145

4.2 General Theory of Bundles

Before considering the discrete analogue of connections on principal bundles, we will review some

basic material on the general theory of bundles, fiber bundles, and principal fiber bundles. A more

in-depth discussion of fiber bundles can be found in Steenrod [1951] and Kobayashi and Nomizu

[1963].

A bundle Q consists of a triple (Q,S, π), where Q and S are topological spaces, respectively

referred to as the bundle space and the base space, and π : Q → S is a continuous map called

the projection. We may assume, without loss of generality, that π is surjective, by considering the

bundle over the image π(Q) ⊂ S.

The fiber over the point x ∈ S, denoted Fx, is given by, Fx = π−1(x). In most situations of

practical interest, the fiber at every point is homeomorphic to a common space F , in which case,

F is the fiber of the bundle, and the bundle is a fiber bundle. The geometry of a fiber bundle is


Fx Q

π

Sx

Figure 4.7: Geometry of a fiber bundle.

A bundle (Q,S, π) is a G-bundle if G acts on Q by left translation, and it is isomorphic to

(Q,Q/G, πQ/G), where Q/G is the orbit space of the G action on Q, and πQ/G is the natural

projection.

If G acts freely on Q, then (Q,S, π) is called a principal G-bundle, or principal bundle, and

G is its structure group. G acting freely on Q implies that each orbit is homeomorphic to G, and

therefore, Q is a fiber bundle with fiber G.

To make the setting for the rest of this chapter more precise, we will adopt the following definition

of a principal bundle,

Definition 4.1. A principal bundle is a manifold Q with a free left action, ρ : G×Q→ Q, of a

Lie group G, such that the natural projection, π : Q→ Q/G, is a submersion. The base space Q/G

146

is often referred to as the shape space S, which is a terminology originating from reduction theory.

We will now consider a few standard techniques for combining bundles together to form new

bundles. These methods include the fiber product, Whitney sum, and the associated bundle con-

struction.

Fiber Product. Given two bundles with the same base space, we can construct a new bundle,

referred to as the fiber product, which has the same base space, and a fiber which is the direct

product of the fibers of the original two bundles. More formally, we have,

Definition 4.2. Given two bundles πi : Qi → S, i = 1, 2, the fiber product is the bundle,

π1 ×S π2 : Q1 ×S Q2 → S,

where Q1 ×S Q2 is the set of all elements (q1, q2) ∈ Q1 × Q2 such that π1(q1) = π2(q2), and the

projection π1 ×S π2 is naturally defined by π1 ×S π2(q1, q2) = π1(q1) = π2(q2). The fiber is given by

(π1 ×Q π2)−1(x) = π−11 (x)× π−1

2 (x).

Whitney Sum. The Whitney sum combines two vector bundles using the fiber product con-

struction.

Definition 4.3. Given two vector bundles τi : Vi → Q, i = 1, 2, with the same base, their Whitney

sum is their fiber product, and it is a vector bundle over Q, and is denoted V1 ⊕ V2. This bundle is

obtained by taking the fiberwise direct sum of the fibers of V1 and V2.

Associated Bundle. Given a principal bundle, π : Q→ Q/G, and a left action, ρ : G×M →M ,

of the Lie group G on a manifold M , we can construct the associated bundle.

Definition 4.4. An associated bundle M with standard fiber M is,

M = Q×GM = (Q×M)/G,

where the action of G on Q ×M is given by g(q,m) = (gq, gm). The class (or orbit) of (q,m) is

denoted [q,m]G or simply [q,m]. The projection πM : Q×GM → Q/G is given by,

πM : ([q,m]G) = π(q) ,

and it is easy to check that it is well-defined and is a surjective submersion.

147

4.3 Connections and Bundles

Before formally introducing the precise definition of a connection, we will attempt to develop some

intuition and motivation for the concept. As alluded to in the introduction to this chapter, a

connection describes the curvature of a space. In the classical Riemannian setting used by Einstein

in his theory of general relativity, the curvature of the space is constructed out of the connection, in

terms of the Christoffel symbols that encode the connection in coordinates.

In the context of principal bundles, the connection provides a means of decomposing the tangent

space to the bundle into complementary spaces, as show in Figure 4.8. Directions in the bundle that

project to zero on the base space are called vertical directions, and a connection specifies a set

of directions, called horizontal directions, at each point, which complements the space of vertical

directions.

vertical directionhorizontal direction

geometric phase

bundle projection

bundle

base space

Figure 4.8: Geometric phase and connections.

In the rest of this section, we will formally define connections on principal bundles, and in the

next section, discrete connections will be introduced in a parallel fashion.

Short Exact Sequence. This decomposition of the tangent space TQ into horizontal and vertical

subspaces yields the following short exact sequence of vector bundles over Q,

0 // V Q // TQπ∗ // π∗TS // 0 ,

where V Q is the vertical subspace of TQ, and π∗TS is the pull-back of TS by the projection

π : Q→ S.

148

Atiyah Sequence. When the short exact sequence above is quotiented modulo G, we obtain an

exact sequence of vector bundles over S,

0 // gi // TQ/G

π∗ // TS // 0 ,

which is called the Atiyah sequence (see, for example Atiyah [1957]; Almeida and Molino [1985];

Mackenzie [1995]). Here, g is the adjoint bundle, which is a special case of an associated bundle

(see Definition 4.4). In particular,

g = Q×G g = (Q× g)/G ,

where the action of G on Q × g is given by g(q, ξ) = (gq,Adgξ), and πg : g → S is given by

πg([q, ξ]G) = π(q).

The maps in the Atiyah sequence, i : (Q× g)/G→ TQ/G and π∗ : TQ/G→ TS, are given by

i([q, ξ]G) = [ξQ(q)]G,

and

π∗([vq]G) = Tπ(vq).

Connection 1-form. Given a connection on a principal fiber bundle π : Q → Q/G, we can

represent this as a Lie algebra-valued connection 1-form, A : TQ → g, constructed as follows

(see, for example, Kobayashi and Nomizu [1963]). Given an element of the Lie algebra ξ ∈ g, the

infinitesimal generator map ξ 7→ ξQ yields a linear isomorphism between g and VqQ for each q ∈ Q.

For each vq ∈ TqQ, we define A(vq) to be the unique ξ ∈ g such that ξQ is equal to the vertical

component of vq.

Proposition 4.1. The connection 1-form, A : TQ → g, of a connection satisfies the following

conditions.

1. The 1-form is G-equivariant, that is,

A TLg = Adg A ,

for every g ∈ G, where Ad denotes the adjoint representation of G in g.

2. The 1-form induces a splitting of the Atiyah sequence, that is,

A(ξQ) = ξ ,

149

for every ξ ∈ g.

Conversely, given a g-valued 1-form A on Q satisfying conditions 1 and 2, there is a unique con-

nection in Q whose connection 1-form is A.

Proof. See page 64 of Kobayashi and Nomizu [1963].

Horizontal Lift. The horizontal lift of a vector field X ∈ X(S) is the unique vector field

Xh ∈ X(Q) which is horizontal and which projects onto X, that is, Tπq(Xhq ) = Xπ(q) for all q ∈ Q.

The horizontal lift is in one-to-one correspondence with the choice of a connection on Q, as the

following proposition states.

Proposition 4.2. Given a connection in Q, and a vector field X ∈ X(S), there is a unique horizontal

lift Xh of X. The lift Xh is left-invariant under the action of G. Conversely, every horizontal vector

field Xh on Q that is left-invariant by G is the lift of a vector field X ∈ X(S).

Proof. See page 65 of Kobayashi and Nomizu [1963].

Connection as a Splitting of the Atiyah Sequence. For a review of the basic concepts of

homological algebra, properties of short exact sequences, and splittings, please refer to Appendix A.

Consider the continuous Atiyah sequence,

0 // gi //

oo(π1,A)

___ TQ/Gπ∗ //

oo

Xh

___ TS // 0

We see that the connection 1-form, A : TQ → g, induces a splitting of the continuous Atiyah

sequence, since

(π1,A) i([q, ξ]G) = (π1,A)([ξQ(q)]g) = [q,A(ξQ(q))]G = [q, ξ]G, for all q ∈ Q, ξ ∈ g,

which is to say that (π1,A) i = 1g. Conversely, given a splitting of the continuous Atiyah sequence,

we can extend the map, by equivariance, to yield a connection 1-form.

The horizontal lift also induces a splitting on the continuous Atiyah sequence, since, by definition,

the horizontal lift of a vector field X ∈ X(S) projects onto X, which is to say that π∗ Xh = 1TS .

The horizontal lift and the connection are related by the fact that

1TQ/G = i (π1,A) +Xh π∗,

which is a simple consequence of the fact that the two splittings are part of the following commutative

150

diagram,

0 // g

1g

i //oo(π1,A)

___ TQ/Gπ∗ //

oo

Xh

___

αA

TS

1T S

// 0

0 // gi1 //

ooπ1

___ g⊕ TSπ2 //

ooi2

___ TS // 0

where αA is an isomorphism (see Appendix A). The isomorphism is given in the following lemma.

Lemma 4.3. The map αA : TQ/G→ g⊕ TS defined by

αA([q, q]G) = [q,A(q, q)]G ⊕ Tπ(q, q),

is a well-defined vector bundle isomorphism. The inverse of αA is given by

α−1A ([q, ξ]G ⊕ (x, x)) = [(x, x)hq + ξq]G.

Proof. See page 15 of Cendra et al. [2001].

This lemma, and its higher-order generalization, that identifies T (2)Q/G with T (2)S ×S 2g, is

critical in allowing us to construct the Lagrange–Poincare operator, which is an intrinsic method of

expressing the reduced equations arising from Lagrangian reduction.

In the next section, we will develop the theory of discrete connections on principal bundles in a

parallel fashion to the way we introduced continuous connections.

4.4 Discrete Connections

Discrete variational mechanics is based on the idea of approximating the tangent bundle TQ of

Lagrangian mechanics with the pair groupoid Q×Q. As such, the purpose of a discrete connection

is to decompose the subset of Q×Q that projects to a neighborhood of the diagonal of S × S into

horizontal and vertical spaces.

The reason why we emphasize that the construction is only valid for the subset of Q × Q that

projects to a neighborhood of the diagonal of S × S is that there are topological obstructions to

globalizing the construction to all of Q×Q except in the case that Q is a trivial bundle.

One of the challenges of dealing with the discrete space modelled by the pair groupoid Q × Q

is that it is not a linear space, in contrast to TQ. As we shall see, the standard pair groupoid

composition is not sufficient to make sense of the notion of an element (q0, q1) ∈ Q × Q being the

composition of a horizontal and a vertical element. We will propose a natural notion of composing

an element with a vertical element that makes sense of the horizontal and vertical decomposition.

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In the subsequent sections, we will use the discrete connection to extend the pair groupoid

composition even further, and explore its applications to the notion of curvature in discrete geometry.

Intrinsic Representation of the Tangent Bundle. The intuition underlying our construction

of discrete horizontal and vertical spaces is best developed by considering the intrinsic representation

of the tangent bundle. This representation is obtained by identifying a tangent vector at a point

on the manifold with the equivalence class of curves on the manifold going through the point, such

that the tangent to the curve at the point is given by the tangent vector. This notion is illustrated

in Figure 4.9.

Figure 4.9: Intrinsic representation of the tangent bundle.

Given a vector vq ∈ TQ, we identify it with the family of curves q : R → Q, such that q(0) = q,

and q(0) = v. The equivalence class [ · ] identifies curves with the same basepoint, and the same

velocity at the basepoint.

With this representation, it is natural to consider (q0, q1) ∈ Q × Q to be an approximation of

[q(·)] = vq ∈ TQ, in the sense that,

q0 = q(0), q1 = q(h),

for some fixed time step h, and where q(·) is a representative curve corresponding to vq in the

intrinsic representation of the tangent bundle.

4.4.1 Horizontal and Vertical Subspaces of Q × Q

Recall that the vertical subspace at a point q, denoted Vq, is given by

Vq = vq ∈ TQ | π∗(vq) = 0 = ξQ | ξ ∈ g.

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Notice that the vertical space is precisely that subspace of TQ which maps under the lifted projection

map to the embedded copy of S in TS. We proceed in an analogous fashion to define a discrete

vertical subspace at a point q.

The natural discrete analogue of the lifted projection map π∗ is the diagonal action of the

projection map on Q×Q, (π, π) : Q×Q→ Q×Q, where (q0, q1) 7→ (πq0, πq1). This is because

π∗(vq) = π∗([q(·)]) = [π(q(·))].

In the same way that we embed S into TS by the map x 7→ [x] = 0x, S naturally embeds itself

into the diagonal of S × S, x 7→ (x, x) = eS×S , which we recall is the identity subspace of the pair

groupoid.

The alternative description of the vertical space is in terms of the embedding of Q× g into TQ,

by (q, ξ) 7→ ξQ(q), using the infinitesimal generator construction,

ξQ(q) = [exp(ξt)q].

In an analogous fashion, we construct a discrete generator map, which is given in the following

definition.

Definition 4.5. The discrete generator is the map i : Q×G→ Q×Q, given by

i(q, g) = (q, gq),

which we also denote by iq(g) = i(q, g) = (q, gq).

Then, we have the following definition of the discrete vertical space.

Definition 4.6. The discrete vertical space is given by

Verq = (q, q′) ∈ Q×Q | (π, π)(q, q′) = eS×S

= iq(g) | g ∈ G.

This is the discrete analogue of the statement Verq = vq ∈ TQ | π∗(vq) = 0 = ξQ | ξ ∈ g.

Since the pair groupoid composition is only defined on the space of composable pairs, we need

to extend the composition to make sense of how the discrete horizontal space is complementary to

the discrete vertical space. In particular, we define the composition of a vertical element with an

arbitrary element of Q×Q as follows.

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Definition 4.7. The composition of an arbitrary element (q0, q1) ∈ Q × Q with a vertical element

is given by

iq0(g) · (q0, q1) = (e, g)(q0, q1) = (q0, gq1).

An elementary consequence of this definition is that it makes the discrete generator map a

homomorphism.

Lemma 4.4. The discrete generator, iq, is a homomorphism. This is a discrete analogue of the

statement in the continuous theory that (ξ + χ)Q = ξQ + χQ.

Proof. We compute,

iq(g) · iq(h) = iq(g) · (q, hq)

= (e, g)(q, hq)

= (q, ghq)

= iq(gh).

Therefore, iq is a homomorphism.

If we define the G action on Q×G to be h(q, g) = (hq, hgh−1), we find that the composition of

a vertical element with an arbitrary element is G-equivariant.

Lemma 4.5. The composition of a vertical element with an arbitrary element of Q × Q is G-

equivariant,

ihq0(hgh−1) · (hq0, hq1) = h · iq0(g) · (q0, q1) .

Proof. Consider the following computation,

ihq0(hgh−1) · (hq0, hq1) = (hq0, hgh−1hq1)

= (hq0, hgq1)

= h(q0, gq1)

= h · iq0(g) · (q0, q1) .

Having made sense of how to compose an arbitrary element of Q×Q with a vertical element, we

are in a position to introduce the notion of a discrete connection.

A discrete connection is a G-equivariant choice of a subset of Q × Q called the discrete

horizontal space, that is complementary to the discrete vertical space. In particular, given

(q0, q1) ∈ Q×Q, a discrete connection decomposes this into the horizontal component, hor(q0, q1),

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and the vertical component, ver(q0, q1), such that

ver(q0, q1) · hor(q0, q1) = (q0, q1) ,

in the sense of the composition of a vertical element with an arbitrary element we defined previously.

Furthermore, the G-equivariance condition states that

hor(gq0, gq1) = g · hor(q0, q1) ,

and

ver(gq0, gq1) = g · ver(q0, q1) .

4.4.2 Discrete Atiyah Sequence

Recall that we obtain a short exact sequence corresponding to the decomposition of TQ into hor-

izontal and vertical spaces. Due to the equivariant nature of the decomposition, quotienting this

short exact sequence yields the Atiyah sequence. In this subsection, we will introduce the analogous

discrete objects.

Short Exact Sequence. The decomposition of the pair groupoid Q×Q, into discrete horizontal

and vertical spaces, yields the following short exact sequence of bundles over Q.

0 // VerQ i // Q×Q(π,π)

// (π, π)∗S × S // 0,

where VerQ is the discrete vertical subspace of Q × Q, and (π, π)∗S × S is the pull-back of S × S

by the projection (π, π) : Q×Q→ S × S.

Discrete Atiyah Sequence. When the short exact sequence above is quotiented modulo G, we

obtain an exact sequence of bundles over S,

0 // Gi // (Q×Q)/G

(π,π)// S × S // 0 ,

which we call the discrete Atiyah sequence. Here, G is an associated bundle (see Definition 4.4).

In particular,

G = Q×G G = (Q×G)/G ,

155

where the action of G on Q × G is given by g(q, h) = (gq, ghg−1), which is the natural discrete

analogue of the adjoint action of g on Q×g. Furthermore, πG : G→ S is given by πG([q, g]G) = π(q).

The maps in the discrete Atiyah sequence i : G→ (Q×Q)/G, and (π, π) : (Q×Q)/G] → S×S,

are given by

i([q, g]G) = [q, gq]G = [iq(g)]G ,

and

(π, π)([q0, q1]g) = (πq0, πq1) .

4.4.3 Equivalent Representations of a Discrete Connection

In addition to the discrete connection which arises from a G-equivariant decomposition of the pair

groupoid Q×Q into a discrete horizontal and vertical space, we have equivalent representations in

terms of splittings of the discrete Atiyah sequence, as well as maps on the unreduced short exact

sequence.

Maps on the Unreduced Short Exact Sequence. These correspond to discrete analogues of

the connection 1-form, and the horizontal lift.

• Discrete connection 1-form, Ad : Q×Q→ G.

• Discrete horizontal lift, (·, ·)hq : S ×Q→ Q×Q.

Maps That Yield a Splitting of the Discrete Atiyah Sequence.

• (π1,Ad) : (Q×Q)/G→ G, which is related to the discrete connection 1-form.

• (·, ·)h : S × S → (Q×Q)/G, which is related to the discrete horizontal lift.

Relating the Two Sets of Representations. These two sets of representations are related in

the following way:

• The maps on the unreduced short exact sequence are equivariant, and hence drop to the

discrete Atiyah sequence, where they induce splittings of the short exact sequence.

• The maps that yield splittings of the discrete Atiyah sequence can be extended equivariantly

to recover the maps on the unreduced short exact sequence.

Furthermore, standard results from homological algebra (see Appendix A) yield an equivalence

between the two splittings of the discrete Atiyah sequence.

156

In the rest of this section, we will also discuss in detail the method of moving between the

various representations of the discrete connection. The organization of the rest of the section, and

the subsections in which we relate the various representations are given in the following diagram.

§4.4.1discrete connectionhor : Q×Q→ Horqver : Q×Q→ Verq

§4.4.4discrete connection 1-form

Ad : Q×Q→ G

zz

§4.4.4::uuuuuuuuuuu

§4.4.5discrete horizontal lift(·, ·)hq : S × S → Q×Q

$$

§4.4.5ddIIIIIIIIIII

//§4.4.5

oo

§4.4.6splitting (connection 1-form)

(π,Ad) : (Q×Q)/G→ G

§4.4.6

OO

§4.4.7splitting (horizontal lift)

(·, ·)h : S × S → (Q×Q)/G

§4.4.7

OO

4.4.4 Discrete Connection 1-Form

Given a discrete connection on a principal fiber bundle π : Q→ Q/G, we can represent this as a Lie

group-valued discrete connection 1-form, Ad : Q×Q→ G, which is a natural generalization of

the Lie algebra-valued connection 1-form on tangent bundles, A : TQ→ g, to the discrete context.

Discrete Connection 1-Forms from Discrete Connections. The discrete connection 1-form

is constructed as follows. Given an element of the Lie group g ∈ G, the discrete generator map

g 7→ iq(g) yields an isomorphism between G and Verq for each q ∈ Q. For each (q0, q1) ∈ Q × Q,

we define Ad(q0, q1) to be the unique g ∈ G such that iq(g) is equal to the vertical component of

(q0, q1). In particular, this is equivalent to the condition that the following statement holds,

(q0, q1) = iq0(Ad(q0, q1)) · hor(q0, q1).

Remark 4.1. It follows from the above identity that the discrete horizontal space can also be ex-

pressed as

Horq0 = (q0, q1) ∈ Q×Q | hor(q0, q1) = (q0, q1)

= (q0, q1) ∈ Q×Q | Ad(q0, q1) = e .

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We will now establish a few properties of the discrete connection 1-form.

Proposition 4.6. The discrete connection 1-form, Ad : Q × Q → G, satisfies the following

properties.

1. The 1-form is G-equivariant, that is,

Ad Lg = Ig Ad,

which is the discrete analogue of the G-equivariance of the continuous connection, A TLg =

Adg A.

2. The 1-form induces a splitting of the Discrete Atiyah sequence, that is,

Ad(iq(g)) = Ad(q0, gq0) = g,

which is the discrete analogue of A(ξQ) = ξ.

Proof. The proof relies on the properties of a discrete connection, and the definition of the discrete

connection 1-form.

1. The discrete connection 1-form satisfies the condition

(q0, q1) = iq0(Ad(q0, q1)) · hor(q0, q1) .

If we denote hor(q0, q1) by (q0, q1), we have that

(q0, q1) = (q0,Ad(q0, q1)q1) .

Similarly, we have,

(gq0, gq1) = igq0(Ad(gq0, gq1)) · hor(gq0, gq1)

= igq0(Ad(gq0, gq1)) · g · hor(q0, q1)

= (e,Ad(gq0, gq1))(gq0, gq1)

= (gq0,Ad(gq0, gq1)gq1) ,

where we have used the G-equivariance of the discrete horizontal space. By looking at the

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expressions for gq1 and q1, we conclude that

Ad(gq0, gq1)gq1 = gq1

= gAd(q0, q1)q1 ,

Ad(gq0, gq1)g = gAd(q0, q1) ,

Ad(gq0, gq1) = gAd(q0, q1)g−1 ,

which is precisely the statement that Ad Lg = Ig Ad, that is to say that Ad is G-equivariant.

2. Recall that iq(g) is an element of the discrete vertical space. Since the discrete horizontal

space is complementary to the discrete vertical space, it follows that ver(iq(g)) = iq(g). Then,

by the construction of the discrete connection 1-form, Ad(iq(g)) is the unique element of G

such that

iq(Ad(iq(g))) = ver(iq(g)) = iq(g) .

Since iq is an isomorphism between G and the discrete vertical space, we conclude that

Ad(iq(g)) = g, as desired.

The second result is equivalent to the map recovering the discrete Euler–Poincare connection

when restricted to a G-fiber, that is, Ad(x, g0, x, g1) = g1g−10 . In particular, it follows that the map

is trivial when restricted to the diagonal space, that is, Ad(q, q) = e.

The properties of a discrete connection are discrete analogues of the properties of a continuous

connection in the sense that if a discrete connection has a given property, the corresponding contin-

uous connection which is induced in the infinitesimal limit has the analogous continuous property.

The precise sense in which a discrete connection induces a continuous connection will be discussed

in §4.5.2.

Discrete Connections from Discrete Connection 1-Forms. Having shown how to obtain a

discrete connection 1-form from a discrete connection, let us consider the converse case of obtaining

a discrete connection from a discrete connection 1-form with the properties above. We do this by

constructing the discrete horizontal and vertical components as follows.

Definition 4.8. Given a discrete connection 1-form, Ad : Q × Q → G that is G-equivariant and

induces a splitting of the discrete Atiyah sequence, we define the horizontal component to be

hor(q0, q1) = iq0((Ad(q0, q1))−1) · (q0, q1) .

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The vertical component is given by

ver(q0, q1) = iq0(Ad(q0, q1)) .

Proposition 4.7. The discrete connection we obtain from a discrete connection 1-form has the

following properties.

1. The discrete connection yields a horizontal and vertical decomposition of Q×Q, in the sense

that

(q0, q1) = ver(q0, q1) · hor(q0, q1) ,

for all (q0, q1) ∈ Q×Q.

2. The discrete connection is G-equivariant, in the sense that

hor(gq0, gq1) = g · hor(q0, q1) ,

and

ver(gq0, gq1) = g · ver(q0, q1) .

Proof. The proof relies on the properties of the discrete connection 1-form, and the definitions of

the discrete horizontal and vertical spaces.

1. Consider the following computation,

ver(q0, q1) · hor(q0, q1) = iq0(Ad(q0, q1)) · iq0((Ad(q0, q1))−1) · (q0, q1)

= iq0(Ad(q0, q1)(Ad(q0, q1))−1) · (q0, q1)

= iq0(e) · (q0, q1)

= (q0, q1) ,

where we used that iq is a homomorphism (see Lemma 4.4).

2. We compute,

hor(gq0, gq1) = igq0((Ad(gq0, gq1))−1) · (gq0, gq1)

= igq0(g(Ad(q0, q1))−1g−1) · (gq0, gq1)

= (e, g(Ad(q0, q1))−1g−1)(gq0, gq1)

160

= (gq0, g(Ad(q0, q1))−1q1)

= g · iq0((Ad(q0, q1))−1) · (q0, q1)

= g · hor(q0, q1) ,

where we have used the fact that the composition of a vertical element with an arbitrary

element is G-equivariant (see Lemma 4.5). Similarly, we compute,

ver(gq0, gq1) = igq0(Ad(gq0, gq1))

= igq0(gAd(q0, q1)g−1)

= (gq0, gAd(q0, q1)g−1gq0)

= g · (q0,Ad(q0, q1)q0)

= g · iq0(Ad(q0, q1))

= g · ver(q0, q1) .

Local Representation of the Discrete Connection 1-Form. Since the discrete connection

1-form can be thought of as comparing group fiber quantities at different base points, we obtain the

natural identity that

Ad(gq0, hq1) = hAd(q0, q1)g−1.

In a local trivialization, this corresponds to

Ad(x0, g0, x1, g1) = g1Ad(x0, e, x1, e)g−10 .

We define

A(x0, x1) = Ad(x0, e, x1, e),

which yields the local representation of the discrete connection 1-form.

Definition 4.9. Given a discrete connection 1-form, Ad : Q × Q → G, its local representation

is given by

Ad(x0, g0, x1, g1) = g1A(x0, x1)g−10 ,

where

A(x0, x1) = Ad(x0, e, x1, e) .

Lemma 4.8. The local representation of a discrete connection is G-equivariant.

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Proof. Consider the following computation,

Ad(g(x0, g0), g(x1, g1)) = Ad((x0, gg0), (x1, gg1))

= gg1A(x0, x1)(gg0)−1

= g(g1A(x0, x1)g−10 )g−1

= gAd((x0, g0), (x1, g1))g−1 ,

which shows that the local representation is G-equivariant, as expected.

Notice also that in the pure group case, where Q = G, this recovers the discrete Euler–Poincare

connection, as we would expect, since the shape space is trivial. In particular, x0 = x1 = e, which

implies that A(x0, x1) = Ad(e, e, e, e) = e, and Ad(g0, g1) = g1g−10 .

Example 4.1. As an example, we construct the natural discrete analogue of the mechanical con-

nection, A : TQ → g, by the following procedure, which yields a discrete connection 1-form,

Ad : Q×Q→ G.

1. Given the point (q0, q1) ∈ Q × Q, we construct the geodesic path q01 : [0, 1] → Q with respect

to the kinetic energy metric, such that q01(0) = q0, and q01(1) = q1.

2. Project the geodesic path to the shape space, x01(t) ≡ πq01(t), to obtain the curve x01 on S.

3. Taking the horizontal lift of x01 to Q using the connection A yields q01.

4. There is a unique g ∈ G such that q01(1) = g · q01(1).

5. Define Ad(q0, q1) = g.

This discrete connection is consistent with the classical notion of a connection in the limit that q1

approaches q0, in the usual sense in which discrete mechanics on Q × Q converges to continuous

Lagrangian mechanics on TQ. As mentioned before, this statement is made more precise in §4.5.2.

4.4.5 Discrete Horizontal Lift

The discrete horizontal lift of an element (x0, x1) ∈ S × S is the subset of Q × Q that are

horizontal elements, and project to (x0, x1). Once we specify the base point q ∈ Q, the discrete

horizontal lift is unique, and we introduce the map (·, ·)hq : S × S → Q×Q.

Discrete Horizontal Lifts from Discrete Connections. The discrete horizontal lift can be

constructed once the discrete horizontal space is defined by a choice of discrete connection.

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Definition 4.10. The discrete horizontal lift is the unique map (·, ·)hq : S × S → Q × Q, such

that

(π, π) · (x0, x1)hq = (x0, x1) ,

and

(x0, x1)hq ∈ Horq .

Lemma 4.9. The discrete horizontal lift is G-equivariant, which is to say that

(x0, x1)hgq = g · (x0, x1)hq .

Proof. Given (x0, x1) ∈ S × S, denote (x0, x1)hq0 by (q0, q1). Then, by the definition of the discrete

horizontal lift, we have that

(π, π) · (q0, q1) = (x0, x1) ,

and it follows that

(π, π) · (gq0, gq1) = (x0, x1) .

Also, from the definition of the discrete horizontal lift,

(q0, q1) ∈ Horq0 ,

and by the G-equivariance of the horizontal space,

(gq0, gq1) ∈ g ·Horq0 = Horgq0 .

This implies that (gq0, gq1) satisfies the conditions for being the discrete horizontal lift of (x0, x1)

with basepoint gq0. Therefore, (x0, x1)hgq0 = (gq0, gq1) = g · (q0, q1) = g · (x0, x1)hq0 , as desired.

Discrete Connections from Discrete Horizontal Lifts. Conversely, given a discrete horizontal

lift, we can recover a discrete connection.

Definition 4.11. Given a discrete horizontal lift, we define the horizontal component to be

hor(q0, q1) = (π(q0, q1))hq0 ,

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and the vertical component is given by

ver(q0, q1) = iq0(g) ,

where g is the unique group element such that

(q0, q1) = iq0(g) · hor(q0, q1) .

The last expression simply states that the discrete horizontal and vertical space are complemen-

tary with respect to the composition we defined between a vertical element and an arbitrary element

of Q×Q.

Discrete Horizontal Lifts from Discrete Connection 1-Forms. We wish to construct a

discrete horizontal lift (·, ·)h : S × S → (Q × Q)/G, given a discrete connection Ad : Q × Q → G.

We state the construction of such a discrete horizontal lift as a proposition.

Proposition 4.10. Given a discrete connection 1-form, Ad : Q × Q → G, the discrete horizontal

lift is given by

(x0, x1)h = [π−1(x0, x1) ∩ A−1d (e)]G.

Furthermore, the discrete horizontal lift satisfies the following identity,

iq0(Ad(q0, q1)) · (π(q0, q1))hq0 = (q0, q1),

which implies that the discrete connection 1-form and the discrete horizontal lift induces a horizontal

and vertical decomposition of Q×Q.

The horizontal lift can be expressed in a local trivialization, where q0 = (x0, g0), using the local

expression for the discrete connection,

(x0, x1)hq0 = (x0, g0, x1, g0(A(x0, x1))−1).

Proof. We will show that this operation is well-defined on the quotient space. Using the local

representation of the discrete connection in the local trivialization (see Definition 4.9), we have,

A−1d (e) ∩ π−1(x0, x1)

= (x0, g, x1, g · (A(x0, x1))−1) | x0, x1 ∈ S, g ∈ G

∩ (x0, h0, x1, h1) | h0, h1 ∈ G

= (x0, g, x1, g · (A(x0, x1))−1) | h ∈ G

164

= G · (x0, e, x1, (A(x0, x1))−1),

which is a well-defined element of (Q × Q)/G. Since this is true in a local trivialization, and both

the discrete connection and projection operators are globally defined, this inverse coset is globally

well-defined as an element of (Q×Q)/G.

In particular, the computation above allows us to obtain a local expression for the discrete

horizontal lift in terms of the local representation of the discrete connection. That is,

(x0, x1)h(x0,e)= (x0, e, x1, (A(x0, x1))−1),

(x0, x1)h = [(x0, e, x1, (A(x0, x1))−1)]G.

By the properties of the discrete horizontal lift, this extends to π−1(x0, x1) ⊂ Q×Q,

(x0, x1)h(x0,g)= (x0, x1)hg(x0,e)

= g · (x0, x1)h(x0,e)

= g(x0, e, x1, (A(x0, x1))−1)

= (x0, g, x1, g(A(x0, x1))−1).

To prove the second claim, we have in the local trivialization of Q×Q, (q0, q1) = (x0, g0, x1, g1).

Then, by the result above,

(π(q0, q1))hq0 = (π(q0, q1))h(x0,g0)= (x0, g0, x1, g0(A(x0, x1))−1).

Also, by the local representation of the discrete connection,

Ad(q0, q1) = g1A(x0, x1)g−10 .

Therefore,

iq0(Ad(q0, q1) · (π(q0, q1))hq0 = (e,Ad(q0, q1)) · (π(q0, q1))hq0

= (e, g1A(x0, x1)g−10 ) · (x0, g0, x1, g0(A(x0, x1))−1)

= (x0, g0, x1, (g1A(x0, x1)g−10 )(g0(A(x0, x1))−1))

= (x0, g0, x1, g1)

= (q0, q1),

165

as claimed.

Discrete Connection 1-Forms from Discrete Horizontal Lifts. Given a horizontal lift (·, ·)hq :

S × S → Q×Q, we wish to construct a discrete connection 1-form, Ad : Q×Q→ G.

Lemma 4.11. Given a discrete horizontal lift, (·, ·)hq : S × S → Q ×Q, the discrete connection

1-form, Ad : Q×Q→ G, is uniquely defined by the following identity,

iq0(Ad(q0, q1)) · (π(q0, q1))hq0 = (q0, q1).

Proof. To show that this construction is well-defined, we note that π1(q0, q1) = π1(π(q0, q1))hq0 , by the

construction of (·, ·)hq0 from (·, ·)h : S × S → (Q×Q)/G. Furthermore, π2(q0, q1) and π2(π(q0, q1))hq0are in the same fiber of the principal bundle π : Q → Q/G and are therefore related by a unique

element g ∈ G. Since this element is unique, Ad(q0, q1) is uniquely defined by the identity.

4.4.6 Splitting of the Discrete Atiyah Sequence (Connection 1-Form)

Consider the discrete Atiyah sequence,

0 // G(q,gq)

//oo(π1,Ad)

___ (Q×Q)/G(π,π)

//oo

(·,·)h

___ S × S // 0 .

As we see from Theorem A.2, given a short exact sequence

0 // A1

f//

ook

___ Bg

//oo

h___ A2

// 0 ,

there are three equivalent conditions under which the exact sequence is split. They are as follows,

1. There is a homomorphism h : A2 → B with g h = 1A2 ;

2. There is a homomorphism k : B → A1 with k f = 1A1 ;

3. The given sequence is isomorphic (with identity maps on A1 and A2) to the direct sum short

exact sequence,

0 // A1i1 // A1 ⊕A2

π2 // A2// 0 ,

and in particular, B ∼= A1 ⊕A2.

We will address all three conditions in this and the next two subsections.

166

Splittings from Discrete Connection 1-Forms. A discrete connection 1-form, Ad : Q×Q→ G,

induces a splitting of the discrete Atiyah sequence, in the sense that

(π1,Ad) i = 1G .

Lemma 4.12. Given a discrete connection 1-form, Ad : Q × Q → G, we obtain a splitting of the

discrete Atiyah sequence, ϕ : (Q×Q)/G→ G, which is given by

ϕ([q0, q1]G) = [q0,Ad(q0, q1)]G.

We denote this map by (π1,Ad).

Proof. This expression is well-defined, as the following computation shows,

ϕ([gq0, gq1]G) = [gq0,Ad(gq0, gq1)]G

= [gq0, gAd(q0, q1)g−1]G

= ϕ([q0, q1]G).

Furthermore, since

(π1,Ad) i([q, g]G) = (π1,Ad)([q, gq]G)

= [π1(q, gq),Ad(q, gq)]G

= [q, g]G,

it follows that we obtain a splitting of the discrete Atiyah sequence.

Discrete Connection 1-Forms from Splittings. Given a splitting of the discrete Atiyah se-

quence, we can obtain a discrete connection 1-form using the following construction.

Given [q0, q1]G ∈ (Q × Q)/G, we obtain from the splitting of the discrete Atiyah sequence

an element, [q, g]G ∈ G. Viewing [q, g]G as a subset of Q × G, consider the unique g such that

(q0, g) ∈ [q, g]G ⊂ Q×G. Then, we define

Ad(x0, e, x1, g−10 g1) = g.

We extend this definition to the whole of Q×Q by equivariance,

Ad(x0, g0, x1, g1) = g0gg−10 .

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Lemma 4.13. Given a splitting of the discrete Atiyah sequence, the construction above yields a

discrete connection 1-form with the requisite properties.

Proof. To show that the Ad satisfies the properties of a discrete connection 1-form, we first note

that equivariance follows from the construction.

Since we have a splitting, it follows that ϕ([q, gq]G) = [q, g]G, as ϕ composed with the map from

G to (Q×Q)/G is the identity on G. Using a local trivialization, we have,

[q0, g]G = ϕ([q0, gq0]G)

= ϕ([(x0, e), (x0, g−10 gg0)]G)

= [(x0, e), g]G

= [(x0, g0), g0gg−10 ]G.

Then, by definition,

Ad((x0, e), (x0, g−10 gg0)) = g,

and furthermore, g = g0gg−10 . From this, we conclude that

Ad(q0, gq0) = Ad((x0, g0), (x0, gg0))

= g0Ad((x0, e), (x0, g−10 gg0))g−1

0

= g0gg−10

= g.

Therefore, we have that Ad(q0, gq0) = g, which together with equivariance implies that Ad is a

discrete connection 1-form.

4.4.7 Splitting of the Discrete Atiyah Sequence (Horizontal Lift)

As was the case with the discrete connection 1-form, the discrete horizontal lift is in one-to-one

correspondence with splittings of the discrete Atiyah sequence, and they are related by taking the

quotient, or extending by G-equivariance, as appropriate.

Splittings from Discrete Horizontal Lifts. Given a discrete horizontal lift, we obtain a splitting

by taking its quotient.

Lemma 4.14. Given a discrete horizontal lift, (·, ·)hq : S × S → Q × Q, the map (·, ·)h : S × S →

(Q×Q)/G, which is given by

(x0, x1)h = [(x0, x1)h(x0,e)]G ,

168

induces a splitting of the discrete Atiyah sequence.

Proof. We compute,

(π, π) (x0, x1)h = (π, π)([(x0, x1)h(x0,e)]G)

= (x0, x1) ,

where we used the G-equivariance of the discrete horizontal lift, and the property that (π, π) ·

(x0, x1)hq = (x0, x1) for any q ∈ Q. This implies that (π, π) (·, ·)h = 1S×S , as desired.

Discrete Horizontal Lifts from Splittings. Given a splitting, (·, ·)h : S × S → (Q×Q)/G, we

obtain a discrete horizontal lift, (·, ·)hq : S × S → Q×Q, using the following construction.

We denote by (x0, x1)hq0 the unique element in (x0, x1)h, thought of as a subset of Q ×Q, such

that the first component is q0. This is the discrete horizontal lift of the point (x0, x1) ∈ S×S where

the base point is specified.

Lemma 4.15. Given a splitting of the discrete Atiyah sequence, the construction above yields a

discrete horizontal lift with the requisite properties.

Proof. Since the quotient space (Q × Q)/G is obtained by the diagonal action of G on Q × Q, it

follows that if (x0, x1)hq0 ∈ (x0, x1)h ⊂ Q×Q, then g · (x0, x1)hq0 ∈ (x0, x1)h ⊂ Q×Q. Since the first

component of G · (x0, x1)hq0 is gq0, and g · (x0, x1)hq0 ∈ (x0, x1)h ⊂ Q×Q, we have that

(x0, x1)hgq0 = g · (x0, x1)hq0 ,

which is to say that the discrete horizontal lift constructed above is G-equivariant.

Since (·, ·)h is a splitting of the discrete Atiyah sequence, we have that (π, π) (·, ·)h = 1S×S ,

and this implies that any element in (x0, x1)h, viewed as a subset of Q × Q, projects to (x0, x1).

Therefore, the discrete horizontal lift we constructed above has the requisite properties.

4.4.8 Isomorphism between (Q × Q)/G and (S × S) ⊕ G

The notion of a discrete connection is motivated by the desire to construct a global diffeomorphism

between (Q × Q)/G → S and (S × S) ⊕ G → S. This is the discrete analogue of the identification

between TQ/G→ Q/G and T (Q/G)⊕ g → Q/G which is the context for Lagrangian Reduction in

Cendra et al. [2001]. Since a choice of discrete connection corresponds to a choice of splitting of the

discrete Atiyah sequence, we have the following commutative diagram, where each row is a short

169

exact sequence.

0 // G

1G

(q,gq)//

oo(π1,Ad)

___ (Q×Q)/G(π,π)

//oo

(·,·)h

___

αAd

S × S

1S×S

// 0

0 // Gi1 //

ooπ1

___ G⊕ (S × S)π2 //

ooi2

___ S × S // 0

Here, we see how the identification between (Q×Q)/G and (S×S)⊕ G are naturally related to the

discrete connection and the discrete horizontal lift.

Recall that the discrete adjoint bundle G is the associated bundle one obtains when M = G, and

ρg acts by conjugation. Furthermore, the action of G on Q × Q is by the diagonal action, and the

action of G×G on Q×Q is component-wise.

Proposition 4.16. The map αAd: (Q×Q)/G→ (S × S)⊕ G defined by

αAd([q0, q1]G) = (πq0, πq1)⊕ [q0,Ad(q0, q1)]G,

is a well-defined bundle isomorphism. The inverse of αAdis given by

α−1Ad

((x0, x1)⊕ [q, g]G) = [(e, g) · (x0, x1)hq ]G,

for any q ∈ Q such that πq = x0.

Proof. To show that αAdis well-defined, note that for any g ∈ G, we have that

(πgq0, πgq1) = (πq0, πq1),

and also,

[gq0,Ad(gq0, gq1)]G = [gq0, gAd(q0, q1)g−1]G

= [q0,Ad(q0, q1)]G.

Then, we see that

αAd([gq0, gq1]G) = αAd

([q0, q1]G).

To show that α−1Ad

is well-defined, note that for any k ∈ G,

(x0, x1)hkq = k · (x0, x1)hq ,

170

and that

α−1Ad

((x0, x1)⊕ [kq, kgk−1]G) = [(e, kgk−1) · (x0, x1)hkq]G

= [(e, kgk−1) · k · (x0, x1)hq ]G

= [(ek, kgk−1k) · (x0, x1)hq ]G

= [(ke, kg) · (x0, x1)hq ]G

= [k · (e, g) · (x0, x1)hq ]G

= [(e, g) · (x0, x1)hq ]G

= α−1Ad

((x0, x1)⊕ [q, g]G).

Example 4.2. It is illustrative to consider the notion of a discrete connection, and the isomorphism,

in the degenerate case when Q = G, which is the context of discrete Euler–Poincare reduction. Here,

the isomorphism is between (G×G)/G and G, and the connection Ad : G×G→ G is given by

Ad(g0, g1) = g1 · g−10 .

Then, we have that

αAd([g0, g1]G) = (πg0, πg1)⊕ [g0,Ad(q0, q1)]G

= (e, e)⊕ [g0, g1g−10 ]G .

Taking the inverse, we have,

α−1Ad

([g0, g1g−10 ]G) = [(e, g1g−1

0 ] · (e, e)hg0 ]G

= [(e, g1g−10 ] · (g0, g0)]G

= [eg0, g1g−10 g0]G

= [g0, g1]G ,

as expected.

4.4.9 Discrete Horizontal and Vertical Subspaces Revisited

Having now fully introduces all the equivalent representations of a discrete connection, we can

revisit the notion of discrete horizontal and vertical subspaces in light of the new structures we have

introduced.

171

Consider the following split exact sequence,

0 // A1

f//

ook

___ Bg

//oo

h___ A2

// 0 .

We can decompose any element in B into a A1 and A2 term by considering the following isomorphism,

B ∼= f k(B)⊕ h g(B).

Similarly, in the discrete Atiyah sequence, we can decompose an element of (Q × Q)/G into a

horizontal and vertical piece by performing the analogous construction on the split exact sequence

0 // G(q,gq)

//oo(π1,Ad)

___ (Q×Q)/G(π,π)

//oo

(·,·)h

___ S × S // 0 .

This allows us to define horizontal and vertical spaces associated with the pair groupoid Q×Q, in

terms of all the structures we have introduced.

Definition 4.12. The horizontal space is given by

Horq = (q, q′) ∈ Q×Q | Ad(q, q′) = e

= (πq, x1)hq ∈ Q×Q | x1 ∈ S.

This is the discrete analogue of the statement Horq = vq ∈ TQ | A(vq) = 0 = (vπq)hq ∈ TQ |

vπq ∈ TS.

Definition 4.13. The vertical space is given by

Verq = (q, q′) ∈ Q×Q | (π, π)(q, q′) = eS×S

= iq(g) | g ∈ G.

This is the discrete analogue of the statement Verq = vq ∈ TQ | π∗(vq) = 0 = ξQ | ξ ∈ g.

In particular, we can decompose an element of Q×Q into a horizontal and vertical component.

Definition 4.14. The horizontal component of (q0, q1) ∈ Q×Q is given by

hor(q0, q1) = ((·, ·)h (π, π))(q0, q1) = (πq0, πq1)hq0 .

Definition 4.15. The vertical component of (q0, q1) ∈ Q×Q is given by

ver(q0, q1) = (i (π1,Ad))(q0, q1) = (q0,Ad(q0, q1)q0) = iq0(Ad(q0, q1)).

172

Lemma 4.17. The horizontal component can be expressed as

hor(q0, q1) = iq0((Ad(q0, q1))−1) · (q0, q1).

Proof.

iq0(Ad(q0, q1)−1) · (q0, q1) = (q0, (Ad(q0, q1))−1q0) · (q0, q1)

= (e, (Ad(q0, q1))−1)(q0, q1)

= (q0, (Ad(q0, q1))−1q1).

Clearly, (π, π)(q0, (Ad(q0, q1))−1q1) = (π, π)(q0, q1). Furthermore,

Ad(q0, (Ad(q0, q1))−1q1) = (Ad(q0, q1))−1Ad(q0, q1) = e.

Therefore, by definition, (q0, (Ad(q0, q1))−1q1) = hor(q0, q1).

Lemma 4.18. The horizontal and vertical operators satisfy the following identity,

ver(q0, q1) · hor(q0, q1) = (q0, q1).

Proof.

ver(q0, q1) · hor(q0, q1) = iq0(Ad(q0, q1)) · (iq0((Ad(q0, q1))−1) · (q0, q1))

= (e,Ad(q0, q1))(e, (Ad(q0, q1))−1)(q0, q1)

= (e,Ad(q0, q1))(q0, (Ad(q0, q1))−1q1)

= (q0,Ad(q0, q1)(Ad(q0, q1))−1q1)

= (q0, q1),

as desired.

4.5 Geometric Structures Derived from the Discrete Con-

nection

In this section, we will introduce some of the additional geometric structures that can be derived

from a choice of discrete connection. These structures include an extension of the pair groupoid

composition to take into account the principal bundle structure, continuous connections that are a

173

limit of a discrete connection, and higher-order connection-like structures.

4.5.1 Extending the Pair Groupoid Composition

Recall that the composition of a vertical element (q0, gq0) with an element (q0, q1) is given by

(q0, gq0) · (q0, q1) = (q0, gq1).

The choice of a discrete connection allows us to further extend the composition, in a manner that

is relevant in describing the curvature of a discrete connection. The decomposition of an element

of Q×Q into a horizontal and vertical piece naturally suggests a generalization of the composition

operation on Q×Q (viewed as a pair groupoid), by using the discrete connection, and the principal

bundle structure of Q.

We wish to define a composition on Q×Q such that the composition of (q0, q1) ·(q0, q1) is defined

whenever πq1 = πq0. Furthermore, we require that the extended composition be consistent with the

vertical composition we introduced previously, as well as the pair groupoid composition, whenever

their domains of definition coincide.

The extended composition is obtained by left translating (q0, q1) by a group element h, such

that q1 = hq0, and then using the pair groupoid composition on (q0, q1) and the left translated term

h(q0, g1). This yields the following intrinsic definition of the extended composition.

Definition 4.16. The extended pair groupoid composition of (q0, q1), (q0, q1) ∈ Q×Q is defined

whenever πq1 = πq0, and it is given by

(q0, q1) · (q0, q1) = (q0,Ad(q0, q1)q1).

As the following lemmas show, this extended composition is consistent with the vertical compo-

sition and the pair groupoid composition.

Lemma 4.19. The extended pair groupoid composition is consistent with the composition of a ver-

tical element with an arbitrary element.

Proof. Consider the composition of a vertical element with an arbitrary element,

(q0, gq0) · (q0, q1) = (q0, gq1).

174

This is consistent with the result using the extended composition,

(q0, gq0) · (q0, q1) = (q0,Ad(q0, gq0)q1)

= (q0, gq1) ,

where we used that the discrete connection yields a splitting of the Atiyah sequence.

Lemma 4.20. The extended pair groupoid composition is consistent with the pair groupoid compo-

sition.

Proof. The pair groupoid composition is given by

(q0, q1) · (q1, q2) = (q0, q2) .

This is consistent with the extended composition,

(q0, q1) · (q1, q2) = (q0,Ad(q1, q1)q2)

= (q0, eq2)

= (q0, q2) .

The extended composition is G-equivariant, and is well-defined on the quotient space, as the

following lemma shows.

Lemma 4.21. The composition · : (Q×Q)× (Q×Q) → (Q×Q) is G-equivariant, that is,

(gq0, gq1) · (gq0, gq1) = g · ((q0, q1) · (q0, q1)).

Furthermore, the composition induces a well-defined quotient composition · : ((Q×Q)×(Q×Q))/G→

(Q×Q)/G.

Proof. Given g ∈ G, we consider,

(gq0, gq1) · (gq0, gq1) = (gq0,Ad(gq0, gq1)gq1)

= (gq0, gAd(q0, q1)g−1gq1)

= (gq0, gAd(q0, q1)q1)

= g · (q0,Ad(q0, q1)q1)

= g · ((q0, q1) · (q0, q1)) ,

175

where we used the equivariance of the discrete connection. It follows that the composition is equiv-

ariant. Furthermore,

[(gq0, gq1) · (gq0, gq1)]G = [(q0, q1) · (q0, q1)]G,

which means that · : ((Q×Q)× (Q×Q))/G→ (Q×Q)/G is well-defined.

Corollary 4.22. The composition of n-terms is G-equivariant. That is to say,

(gq10 , gq11) · (gq20 , gq21) · . . . · (gqn−1

0 , gqn−11 ) · (gqn0 , gqn1 )

= g · ((q10 , q11) · (q20 , q21) · . . . · (qn−10 , qn−1

1 ) · (qn0 , qn1 )).

Proof. The result follows by induction on the previous lemma.

We find that the extended composition we have constructed on the pair groupoid is associative.

However, as we shall see in §4.7.3, composing pair groupoid elements about a loop in the shape space

will not in general yield the identity element eQ×Q, and the defect represents the holonomy about

the loop, which is related to curvature. This may yield the discrete analogue of the expression giving

the geometric phase in terms of a loop integral (in shape space) of the curvature of the connection.

Lemma 4.23. The composition · : (Q×Q)× (Q×Q) → (Q×Q) is associative. That is,

((q00 , q01) · (q10 , q11)) · (q20 , q21) = (q00 , q

01) · ((q10 , q11) · (q20 , q21)) .

Proof. Evaluating the left-hand side, we obtain

((q00 , q01) · (q10 , q11)) · (q20 , q21) = (q00 ,Ad(q10 , q01)q11) · (q20 , q21)

= (q00 ,Ad(q20 ,Ad(q10 , q01)q11)q21)

= (q00 ,Ad(q10 , q01)Ad(q20 , q11)q21) ,

and the right-hand side is given by

(q00 , q01) · ((q10 , q11) · (q20 , q21)) = (q00 , q

01) · (q10 ,Ad(q20 , q11)q21)

= (q00 ,Ad(q10 , q01)Ad(q20 , q11)q21) .

Therefore,

((q00 , q01) · (q10 , q11)) · (q20 , q21) = (q00 , q

01) · ((q10 , q11) · (q20 , q21)) ,

176

and the extended groupoid composition is associative.

4.5.2 Continuous Connections from Discrete Connections

Given a discrete G-valued connection 1-form, Ad : Q × Q → G, we associate to it a continuous

g-valued connection 1-form, A : TQ→ g, by the following construction,

A([q(·)]) = [Ad(q(0), q(·))],

where [·] denotes the equivalence class of curves associated with a tangent vector.

This uses the intrinsic representation of the tangent bundle, which is obtained by identifying a

tangent vector at a point on the manifold with the equivalence class of curves on the manifold going

through the point, such that the tangent to the curve at the point is given by the tangent vector,

which was illustrated earlier in Figure 4.9 on page 151.

More explicitly, given vq ∈ TQ, we consider an associated curve q : [0, 1] → Q, and construct the

curve g : [0, 1] → G, given by

g(t) = Ad(q(0), q(t)).

Then,

A(vq) =d

dt

∣∣∣∣t=0

g(t).

When computing the equations in discrete reduction theory, it is often necessary to consider

horizontal and vertical variations, which we introduce below.

Definition 4.17. We introduce the vertical variation of a point (q0, q1) ∈ Q×Q. Given a curve

qε1 : [0, 1] → Q, such that qε1(0) = q1, the vertical variation is given by

ver δq =d

dε

∣∣∣∣ε=0

ver(q0, qε1) =d

dε

∣∣∣∣ε=0

iq0(Ad(q0, qε1)) .

Definition 4.18. We introduce the horizontal variation of a point (q0, q1) ∈ Q × Q. Given a

curve qε1 : [0, 1] → Q, such that qε1(0) = q1, the horizontal variation is given by

hor δq =d

dε

∣∣∣∣ε=0

hor(q0, qε1) =d

dε

∣∣∣∣ε=0

(π(q0, qε1))hq0 .

4.5.3 Connection-Like Structures on Higher-Order Tangent Bundles

Given a continuous connection, we can construct connection-like structures on higher-order tangent

bundles. This construction is described in detail in Lemma 3.2.1 of Cendra et al. [2001]. In particular,

given a connection 1-form, A : TQ→ g, we obtain a well-defined map, Ak : T (k)Q→ kg.

177

As we will see later, these connection-like structures on higher-order tangent bundles will provide

an intrinsic method of characterizing the order of approximation of a continuous connection by a

discrete connection.

We will describe the discrete analogue of this construction. To begin, the discrete analogue of the

k-th order tangent bundle, T (k)Q, is k + 1 copies of Q, namely Qk+1. Intermediate spaces between

T (k)Q and Qk+1 arise in the general theory of multi-spaces, which is introduced in Olver [2001].

The discrete analogue of tangent lifts, and their higher-order analogues, are obtained by compo-

nentwise application of the map, since a tangent lift of a map is computed by applying the map to a

representative curve, and taking its equivalence class. Therefore, given a map f : M → N , we have

the naturally induced map,

T (k)f : Mk+1 → Nk+1 given by T (k)f(m0, . . . ,mk) = (f(m0), . . . , f(mk)).

And in particular, the group action is lifted to the diagonal group action on the product space.

The discrete connection can be extended to Qk+1 in the natural way, Akd : Qk+1 → ⊕k−1l=0 G ≡ kG,

Akd(q0, . . . , qk) = ⊕k−1l=0 Ad(ql, ql+1).

Similarly, we can define the map from Qk+1 to the Whitney sum of k copies of the conjugate

bundle G by

Qk+1 → kG by (q0, . . . , qk) 7→ ⊕k−1l=0 [q0,Ad(ql, ql+1)]G.

In a natural way, we have the following proposition.

Proposition 4.24. The map

αAkd

: Qk+1 → (Q/G)k+1 ×Q/G kG

defined by

αAkd(q0, . . . , qk) = (πq0, . . . , πqk)×Q/G ⊕k−1

l=0 [q0,Ad(ql, ql+1)]G,

is a well-defined bundle isomorphism. The inverse of αAkd

is given by

α−1Ak

d

((x0, . . . , xk)×Q/G ⊕k−1

l=0 [ql, gl]G)

= [(e, g0, g1g0, . . . , gk−1 . . . g0)) · (x0, . . . , xk)hq0 ]G,

178

where (x0, . . . , xk)hq0 = (q0, . . . , qk) is defined by the conditions:

q0 = q0,

πql = xl,

Ad(ql, ql+1) = e.

Remark 4.2. In a local trivialization, where q0 = (h0, x0), we have,

ql+1 =((A(xl, xl+1) · · ·A(x0, x1))−1h0, xl+1

).

4.6 Computational Aspects

While we saw in the previous section that a discrete connection induces a continuous connection

in the limit, we are often concerned with constructing discrete connections that approximate a

continuous connection to a given order of approximation. This section will address the question of

what it means for a discrete connection to approximate a continuous connection to a given order,

as well as introduce methods for constructing such discrete connections.

4.6.1 Exact Discrete Connection

It is interesting from the point of view of computation to construct an exact discrete connection

associated with a prescribed continuous connection, so that we can make sense of the statement that

a given discrete connection is a k-th order approximation of a continuous connection.

Additional Structure. To construct the exact discrete connection, we require that the configu-

ration manifold Q be endowed with a bi-invariant Riemannian metric, with the property that the

associated exponential map,

exp : TQ→ Q,

is consistent with the group action, in the sense that

exp(ξQ(q)) = exp(ξ) · q.

We extend the exponential to Q×Q as follows,

exp : TQ→ Q×Q,

vq 7→ (q, exp(vq)),

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and denote the inverse by log : Q×Q→ TQ, which is defined in a neighborhood of the diagonal of

Q×Q.

Construction. Having introduced the appropriate structure on the configuration manifold, we

define the exact discrete connection as follows.

Definition 4.19. The Exact Discrete Connection AEd associated with a prescribed continuous

connection A : TQ→ g and a Riemannian metric is given by

AEd (q0, q1) = exp(A(log(q0, q1))).

This construction is more clearly illustrated in the following diagram,

Q×Qlog

//GF ED

AEd

TQA

// gexp

// G

Properties. The exact discrete connection satisfies the properties of the discrete connection 1-

form, in that it is G-equivariant, and it induces a splitting of the discrete Atiyah sequence. The

equivariance of the exact discrete connection arises from the fact that each of the composed maps

is equivariant, and the splitting condition,

AEd (iq(g)) = g,

is a consequence of the compatibility condition,

exp(ξQ(q)) = exp(ξ) · q.

Since the logarithm map is only defined on a neighborhood of the diagonal of Q×Q, it follows that

the exact discrete connection will have the same restriction on its domain of definition.

Example 4.3 (Discrete Mechanical Connection). The continuous mechanical connection is

defined by the following diagram,

T ∗QJ // g∗

TQA

//

FL

OO

g

I

OO

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Correspondingly, the discrete mechanical connection is defined by the following diagram,

Q×QFLd

//GF ED

Jd

T ∗QJ

// g∗

Q×Q@A BCAd

OO// g

exp//

I

OO

G

This is consistent with our notion of an exact discrete mechanical connection as the following diagram

illustrates,

Q×QFLd

//GF ED

Jd

T ∗QJ

// g∗

Q×Q@A BCAd

OO

log// TQ

FL

OO

A // gexp

//

I

OO

G

where the portion in the dotted box recovers the continuous mechanical connection. In checking G-

equivariance, we use the equivariance of exp : g → G, Jd : Q×Q→ g∗, and the equivariance of I in

the sense of a map I : Q→ L(g, g∗), namely, I(gq) ·Adg ξ = Ad∗g−1 I(q) · ξ.

4.6.2 Order of Approximation of a Connection

We have the necessary constructions to consider the order to which a discrete connection approxi-

mates a continuous connection. There are two equivalent ways of defining the order of approximation

of a continuous connection by a discrete connection, the first is more analytical, and is given by the

order of convergence in an appropriate norm on the group.

Definition 4.20 (Order of Connection, Analytic). A discrete connection Ad is a k-th order

discrete connection if, k is the maximum integer for which

∃0 < c <∞,

∃h0 > 0,

such that

supvq ∈ TQ,

|vq| = 1

‖AEd (q, exp(hvq))(Ad(q, exp(hvq)))−1‖ ≤ chk+1, ∀h < h0.

The second definition is more intrinsic, and is related to considering the infinitesimal limit of a

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discrete connection to connection-like structures on higher-order tangent bundles, without the need

for the introduction of the exact discrete connection.

Recall from §4.5.2 that we can construct a continuous connection from a discrete connection by

the following construction,

A([q(·)]) = [Ad(q(0), q(·))].

Given Akd : Qk+1 → kG, we can obtain the continuous limit Ak : T (k)Q→ kg in a similar fashion.

Definition 4.21 (Order of Connection, Intrinsic). A discrete connection Ad is a k-th order

approximation to A if, k is the maximum integer for which the diagram holds,

Ad : Q×Q→ G

A : TQ→ g

Akd : Qk+1 → kG //_______ Ak : T (k)Q→ kg

Here, the double arrows represent the higher-order structures induced by the connections, and the

dotted arrow represents convergence in the limit.

4.6.3 Discrete Connections from Exponentiated Continuous Connections

To apply the exponential and logarithm approach to construct a discrete connection from a pre-

scribed connection, in the sense of the diagram,

Q×Qlog

//GF ED

Ad

TQA

// gexp

// G ,

we can rely on explicit expressions for the exponential and logarithm, such as those given in Ap-

pendix B for the special Euclidean group, or we can rely on approximations to the exponential and

logarithm.

The explicit formulas for the special Euclidean group are particularly useful for applying the

theory of discrete connections to the geometric control of problems such as robotic manipulators, and

clusters of satellites. In dealing with other configuration manifolds, approximants to the exponential

and logarithm may be required due to the absence of explicit formulas.

Even when explicit formulas are available, it may be desirable to rely on a more computationally

efficient approximation, such as the Cayley transformation, methods based on Pade approximants

(see, for example, Cardoso and Silva Leite [2001]; Higham [2001]), or Lie group techniques (see, for

example, Celledoni and Iserles [2000, 2001]; Zanna and Munthe-Kaas [2001/02]). Clearly, these will

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yield a discrete connection that has an order of approximation equal to the lower of the two orders

of approximation of the numerical schemes used for the exponential and the logarithm.

When used in the context of geometric control, high-order approximations to the continuous

connection may not be necessary, since the optimal trajectory is often recomputed at each step, and

in such instances, a low-order approximation suffices.

4.6.4 Discrete Mechanical Connections and Discrete Lagrangians

We will introduce a discrete mechanical connection that is consistent with the structure of discrete

variational mechanics, and will yield a discrete connection that has an order of approximation that

is equation to that obtained from the discrete mechanics.

Consider a G-invariant k-th order discrete Lagrangian, Ld : Q×Q→ R, which is to say that

Ld(gq0, gq1) = Ld(q0, q1),

and

Ld = Lexactd +O(hk+1),

where the exact discrete Lagrangian, Lexactd : Q×Q→ R, is given by

Lexactd (q0, q1) =

∫ h

0

L(q01(t), q01(t))dt.

Here, q01 : [0, h] → Q is the solution of the Euler–Lagrange equations with q01(0) = q0, and

q01(h) = q1. The exact discrete Lagrangian is a generator of the symplectic flow, coming from the

Jacobi solution of the Hamilton–Jacobi equation.

This k-th order discrete Lagrangian yields a k-th order accurate numerical update scheme,

through the discrete Euler–Lagrange equations,

D2Ld(q0, q1) +D1Ld(q1, q2) = 0,

which implicitly defines a discrete flow Φ : (q0, q1) 7→ (q1, q2). By pushing this numerical scheme

forward to T ∗Q using the discrete fiber derivative FLd : Q × Q → T ∗Q, which maps (q0, q1) 7→

(q0,−D1Ld(q0, q1)), we can obtain a Symplectic Partitioned Runge–Kutta scheme of the same order.

We also introduce the discrete momentum map, Jd : Q×Q→ g∗, given by

〈Jd(q0, q1), ξ〉 = −D1Ld(q0, q1) · ξQ(q0).

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The discrete Lagrangian is G-invariant, which implies that for any ξ ∈ g, we have,

Ld(q0, q1) = Ld(exp(ξt) · q0, exp(ξt) · q1),

0 =d

dt

∣∣∣∣t=0

Ld(exp(ξt) · q0, exp(ξt) · q1)

= D1Ld(q0, q1) · ξQ(q0) +D2Ld(q0, q1) · ξQ(q1).

If we restrict to the flow of the discrete Euler–Lagrange equations, we have that

(D1Ld(q0, q1) +D2Ldq(q1, q2)) · ξQ(q1) = 0,

which upon substitution into the previous equation, yields

D1Ld(q0, q1) · ξQ(q0)−D1Ld(q1, q2) · ξQ(q1) = 0,

−D1Ld(q1, q2) · ξQ(q1) = −D1Ld(q0, q1) · ξQ(q0),

〈Jd(q1, q2), ξQ(q1)〉 = 〈Jd(q0, q1), ξQ(q0)〉,

Φ∗Jd = Jd.

which is the statement of the discrete Noether theorem, that the discrete momentum map is preserved

by the discrete Euler–Lagrange flow.

We note that the mechanical connection corresponds to a choice of horizontal space corresponding

to the zero momentum surface. That is to say that the horizontal distribution corresponding to the

continuous mechanical connection is

Horq = vq ∈ TQ | J(vq) = 0,

and the discrete horizontal subspace corresponding to the discrete mechanical connection is

Hordq = (q, q′) ∈ Q×Q | Jd(q, q′) = 0

Remark 4.3. For the discrete horizontal subspace we defined above to have the correct dimension-

ality, the discrete Lagrangian needs to satisfy certain non-degeneracy conditions, which dictates the

size of the neighborhood of the diagonal that the discrete connection is defined on.

Since the continuous momentum map is preserved by the continuous Euler–Lagrange flow, and the

discrete momentum map is preserved by the discrete Euler–Lagrange flow, it follows that the order

of approximation of the continuous mechanical connection by the discrete mechanical connection is

184

equal to the order of approximation of the continuous Euler–Lagrange flow by the discrete Euler–

Lagrange flow. To construct a discrete mechanical connection of a prescribed order, we use the

following procedure.

1. Consider a k-th order G-invariant discrete Lagrangian, Ld : Q×Q→ R,

Ld = Lexactd +O(hk+1).

2. Construct the corresponding discrete momentum map, Jd(q0, q1) → g∗,

〈Jd(q0, q1), ξ〉 = −D1Ld(q0, q1) · ξQ(q0).

3. Then, the k-th order discrete mechanical connection is given implicitly by considering the

condition for the discrete horizontal space,

Ad(q0, q1) = e iff Jd(q0, q1) = 0,

and then extending the construction to the domain of definition by G-equivariance.

4. More explicitly, given (q0, q1) ∈ Q × Q, we consider a local trivialization, in which (q0, q1) =

(x0, g0, x1, g1), and we find the unique g ∈ G such that

Jd(x0, g0, x1, g) = 0.

Then, we have that

Ad(x0, g0, x1, g) = e,

from which we conclude that

Ad(x0, g0, x1, g1) = Ad(x0, g0, x1, g1g−1g)

= g1g−1 · Ad(x0, g0, x1, g)

= g1g−1 .

4.7 Applications

This section will sketch some of the applications of the mathematical machinery of discrete connec-

tions and discrete exterior calculus to problems in computational geometric mechanics, geometric

control theory, and discrete Riemannian geometry.

185

4.7.1 Discrete Lagrangian Reduction

Lagrangian reduction, which is the Lagrangian analogue of Poisson reduction on the Hamiltonian

side, is associated with the reduction of Hamilton’s variational principle for systems with symmetry.

The variation of the action integral associated with a variation in the curve can be expressed in

terms of the Euler–Lagrange operator, EL : T (2)Q → T ∗Q. When the Lagrangian is G-invariant,

the associated Euler–Lagrange operator is G-equivariant, and this induces a reduced Euler–Lagrange

operator, [EL]G : T (2)Q/G → T ∗Q/G. The choice of a connection allows us to construct intrinsic

coordinates on T (2)/G and T ∗Q/G, and the representation of the reduced Euler–Lagrange operator

in these coordinates correspond to the Lagrange–Poincare operator, LP : T (2)(Q/G) ×Q/G 2g →

T ∗(Q/G)⊕ g∗.

The reduced equations obtained by reduction tend to have non-canonical symplectic structures.

As such, naıvely applying standard symplectic algorithms to reduced equations can have undesirable

consequences for the longtime behavior of the simulation, since it preserves the canonical symplectic

form on the reduced space, as opposed to the reduced (non-canonical) symplectic form that is

invariant under the reduced dynamics.

This sends a cautionary message, that it is important to understand the reduction of discrete vari-

ational mechanics, since applying standard numerical algorithms to the reduced equations obtained

from continuous reduction theory may yield undesirable results, inasmuch as long-term stability is

concerned.

Discrete connections on principal bundles provide the appropriate geometric structure to con-

struct a discrete analogue of Lagrangian reduction. We first introduce the discrete Euler–Lagrange

operator, which is constructed as follows.

Discrete Euler–Lagrange Operator. The discrete Euler–Lagrange operator, ELd : Q3 → T ∗Q

satisfies the following property,

dSd(Ld) · δq =∑

ELd(Ld)(qk−1, qk, qk+1) · δqk.

In coordinates, the discrete Euler–Lagrange operator has the form

[D2Ld(qk−1, qk) +D1Ld(qk, qk+1)] dqk.

Discrete Lagrange–Poincare Operator. The map ELd(Ld) : Q3 → T ∗Q, being G-equivariant,

induces a quotient map

[ELd(Ld)]G : Q3/G→ T ∗Q/G,

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which depends only on the reduced discrete Lagrangian ld : (Q × Q)/G → R. We can therefore

identify [ELd(Ld)]G with an operator ELd(ld) which we call the reduced discrete Euler–Lagrange

operator.

If in addition to the principal bundle structure, we have a discrete principal connection as de-

scribed in the previous section, we can identify

Q3/G with (Q/G)3 ×Q/G (G⊕ G).

The isomorphism between these two spaces is a consequence of Proposition 4.24, which is higher-

order generalization of Proposition 4.16. The discrete mechanical connection which was introduced

in §4.6.4 is a particularly natural choice of discrete connection, since it does not require any ad hoc

choices, as it is constructed directly from the discrete Lagrangian.

Furthermore, each discrete G-valued connection 1-form, Ad : Q × Q → G, induces in the in-

finitesimal limit a continuous g-valued connection 1-form, A : TQ → g, as shown in §4.5.2. This

continuous principal connection allows us to identify

T ∗Q/G with T ∗(Q/G)⊕ g∗.

The discrete Lagrange–Poincare operator, LPd(ld) : (Q/G)3 ×Q/G (G⊕ G) → T ∗(Q/G)⊕ g∗, is

obtained from the reduced discrete Euler–Lagrange operator by making the identifications obtained

from the discrete connection structure.

The splitting of the range space of LPd(ld) as a direct product (as in §3.3 of Cendra et al. [2001])

naturally induces a decomposition of the discrete Lagrange–Poincare operator,

LPd(ld) = Hor(LPd(ld))⊕Ver(LPd(ld)),

and this allows the discrete reduced equations to be decomposed in horizontal and vertical equations.

4.7.2 Geometric Control Theory and Formations

There are well-established control algorithms for actuating a control system to achieve a desired

reference configuration. In many problems of practical interest, the actuation of the mechanical

system decomposes into shape and group variables in a natural fashion.

A canonical example of this is a satellite in motion about the Earth, where the orientation of the

satellite is controlled by internal rotors through the use of holonomy and geometric phases, and the

position is controlled by chemical propulsion.

In this example, the configuration space is SE(3), the group is SO(3), and the shape space is

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R3. The group variable corresponds to the orientation, and the shape variable corresponds to the

position. When given an initial and desired configuration, it is desirable, while computing the control

inputs, to decompose the relative motion into a shape component and a group component, so that

they can be individually actuated.

Since the discrete connection is used here to provide an efficient choice of local coordinates for

optimal control, the discrete connection is most naturally obtained by exponentiating the continuous

connection, in the manner described in §4.6.3, in conjunction with the explicit formulas for the expo-

nential and logarithm for SE(3) that are given in Appendix B. The natural choice of the continuous

connection is one in which the horizontal space is given by the momentum surface corresponding to

the current value of the momentum.

To illustrate why it may not be desirable from a control-theoretic point of view to decompose

the space using a trivial connection, consider a satellite that is in a tidally locked orbit about the

Earth, with the initial and desired configuration as illustrated in Figure 4.10.

Initial configuration Desired configuration

Figure 4.10: Application of discrete connections to control.

Here, if we choose a trivial connection, then the relative group element would be a rotation by

π/4, but this choice is undesirable, since the motion is tidally locked, and moving the center of mass

to the new location would result in a shift in the orientation by precisely the desired amount. In

this case, the optimal control input should therefore only actuate in the shape variables, and the

relative group element assigned to this pair of configurations should be the identity element.

As such, the extension of mechanically relevant connections to pairs of points in the configuration

space with finite separation, through the use of a discrete connection, can be of immense value in

geometric control theory.

Similarly, in the case of formations, discrete connections allow for the orientation coordination

problem to be handled in a more efficient manner, by taking into account the dynamic coupling of

the shape and group motions automatically through the use of the discrete mechanical connection.

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4.7.3 Discrete Levi-Civita Connection

Vector bundle connections can be cast in the language of connections on principal bundles by con-

sidering the frame bundle consisting of oriented orthonormal frames over the manifold M , which is a

principal SO(n) bundle, as originally proposed by Cartan [1983, 2001]. For related work on discrete

connections on triangulated manifolds with applications to algebraic topology and the computation

of Chern classes, please see Novikov [2003].

To construct our model of a discrete Riemannian manifold, we first trivialize the frame bundle

to yield SO(n)×M . Then, Q = SO(n)×M , and G = SO(n).

Here, we introduce the notion of a semidiscretized principal bundle, where the shape space, S =

Q/G, is discretized as a simplicial complex K, and the structure group G remains continuous. In this

context, the semidiscretization of the trivialization of the frame bundle is given by Q = SO(n)×K.

A discrete connection is a map Ad : Q × Q → G, and we can construct a candidate for the

Levi-Civita connection on a simplicial complex K, using the discrete analogue of the frame bundle

described above. However, we will first introduce the notion of a discrete Riemannian manifold.

Definition 4.22. A discrete dual Riemannian manifold is a simplicial complex where each

n-simplex σn is endowed with a constant Riemannian metric tensor g, such that the restriction of

the metric tensor to a common face with an adjacent n-simplex is consistent.

This is referred to as a discrete dual Riemannian manifold as we can equivalently think of

associating a Riemannian metric tensor to each dual vertex, and as we shall see, by adopting Cartan’s

method of orthogonal frames (see, for example, Cartan [1983, 2001]), the connection is a SO(n)-

valued discrete dual 1-form, and the curvature is a SO(n)-valued discrete dual 2-form.

For each n-simplex σn, consider an invertible transformation f of Rn such that f∗g = I. In

the orthonormal space, we have a normal operator that maps a (n − 1)-dimensional subspace to a

generator of the orthogonal complement, denoted by ⊥. Then, we obtain a normal operator on the

faces of σn by making the following diagram commute.

f//

⊥

⊥

f//

The coordinate axes in the diagram represent the normalized eigenvectors of the metric, scaled

189

by their respective eigenvalues.

The local representation of the discrete connection is given by

Ad((σn0 , R0), (σn1 , R1)) = R1A(σn0 , σn1 )R−1

0 ,

and so the discrete connection is uniquely defined if we specify A(σn0 , σn1 ), where σn0 and σn1 are

adjacent n-simplices. Since they are adjacent, they share a (n − 1)-simplex, denoted σn−1. In

particular, this can then be thought of as a SO(n)-valued discrete dual 1-form, since to each dual

1-cell, ?σn−1, we associate an element of SO(n).

This element of SO(n) is computed as follows.

1. In each of the n-simplices, we have a normal direction associated with σn−1, denoted by

⊥ (σn−1, σni ) ∈ Rn.

2. If these two normal directions are parallel, we set

〈A, ?σn−1〉 = I,

otherwise, we continue.

3. Construct the (n − 2)-dimensional hyperplane Pn−2, given by the orthogonal complement to

the span of the two normal directions.

Pn−2 =⊥ (span(⊥ (σn−1, σn0 ),⊥ (σn−1, σn1 ))).

4. If ?σn−1 is oriented from σn0 to σn1 , we set

〈A, ?σn−1〉 = R ∈ SO(n) | R|Pn−2 = IPn−2 , R(⊥ (σn−1, σn0 )) =⊥ (σn−1, σn1 ).

The curvature of this discrete Levi-Civita connection is then a SO(n)-valued discrete dual 2-form.

There is however the curious property that the boundary operator for a dual cell complex may not

necessarily agree with the standard notion of boundary, since that may not be expressible in terms

of a chain in the dual cell complex. This is primarily an issue on the boundary of the simplicial

complex, and if we are in the interior, this is not a problem.

Since curvature is a dual 2-form, it is associated with the dual of a codimension-two simplex,

given by ?σn−2. Consider the example illustrated in Figure 4.11.

190

SimplicialComplex,

K

primal(n− 2)-simplex,

σn−2

dual 2-cell,?σn−2

dual 1-chain,∂ ? σn−2

primal(n− 1)-chain,?∂ ? σn−2

Figure 4.11: Discrete curvature as a discrete dual 2-form.

The curvature B of the discrete Levi-Civita connection is given by

〈B, ?σn−2〉 = 〈dA, ?σn−2〉 = 〈A, ∂ ? σn−2〉.

As can be seen from the geometric region ?∂ ? σn−2, the curvature associated with ?σn−2 is given

by the ordered product of the connection associated with the dual cells, ?σn−1, where σn−1 σn−2.

This also suggests that we can also think of the discrete connection as a primal (n − 1)-form, and

the curvature as a primal (n − 2)-form, where the curvature is obtained from the connection using

the codifferential.

When the group G is nonabelian, we see that the curvature is only defined up to conjugation,

since we need to specify a dual vertex on the dual one-chain ∂ ?σn−2 from which to start composing

group elements. To make this well-defined, we can adopt the approach used in defining the simplicial

cup product, and assume that there is a partial ordering on dual vertices, which would make the

curvature unambiguous.

As a quality measure for simplicial triangulations of a Riemannian manifold, having the curvature

defined up to conjugation may be sufficient if we have a norm on SO(n) which is invariant under

conjugation. As an example, taking the logarithm to get an element of the Lie algebra so(n), and

then using a norm on this vector space yields a conjugation-invariant norm on SO(n). This allows

us to detect regions of the mesh with high curvature, and selectively subdivide the triangulation in

such regions.

Similarly, we can define a discrete primal Riemannian manifold, where the Riemannian metric

tensor is associated with primal vertices, and the connection is a G-valued primal 1-form, and the

curvature is a G-valued primal 2-form.

Abstract Simplicial Complex with a Local Metric. Recall that in the §3.4, we introduced

the notion of an abstract simplicial complex with a local metric defined on pairs of vertices that are

adjacent.

191

In this situation, we can compute the curvature around a loop in the mesh using local embed-

dings. We start with an initial n-simplex, which we endow with an orthonormal frame. By locally

embedding adjacent n-simplices into Euclidean space, and parallel transporting the orthonormal

frame, we will eventually transport the frame back to the initial simplex.

The relative orientation between the original frame and the transported frame yields the integral

of the curvature of the surface which is bounded by the traversed curve. This results from a simple

application of the Generalized Stokes’ theorem, and the fact that the curvature is given by the

exterior derivative of the connection 1-form.


We have introduced a complete characterization of discrete connections, in terms of horizontal and

vertical spaces, discrete connection 1-forms, horizontal lifts, and splittings of the discrete Atiyah

sequence. Geometric structures that can be derived from a given discrete connection have been

discussed, including continuous connections, an extended pair groupoid composition, and higher-

order analogues of the discrete connection. In addition, we have explored computational issues, such

as order of accuracy, and the construction of discrete connections from continuous connections.

Applications to discrete reduction theory, geometric control theory, and discrete geometry, have

also been discussed, and it would be desirable to systematically apply the machinery of discrete

connections to these problems.

In addition, connections play a crucial role in representing the nonholonomic constraint distribu-

tion in nonholonomic mechanics, particularly when considering nonholonomic mechanical systems

with symmetry, wherein the nonholonomic connection enters (see, for example, Bloch [2003]). There

has been recent progress on constructing nonholonomic integrators in the work of Cortes [2002] and

McLachlan and Perlmutter [2003], but an intrinsically discrete notion of a connection remains absent

from their work, and they do not consider the role of symmetry reduction in discrete nonholonomic

mechanics.

It would be very interesting to apply the general theory of discrete connections on principal

bundles to nonholonomic mechanical systems with symmetry, and to cast the notion of a discrete

nonholonomic constraint distribution and the nonholonomic connection in the language of discrete

connections, and thereby develop a discrete theory of nonholonomic mechanics with symmetry. This

would be particularly important for the numerical implementation of geometric control algorithms.

The role of discrete connections in the study of discrete geometric phases would also be an area

worth pursuing. A discrete analogue of the rigid-body phase formula, that involves the discrete me-

chanical connection, that is exact for rigid-body simulations that use discrete variational mechanics,

192

would yield significant insights into the geometric structure-preservation properties of variational in-

tegrators. In particular, it would provide much needed insight into how discretization interacts with

geometric phases, and yield an understanding how much of the phase drift observed in a numerical

simulation is due to the underlying geometry of the mechanical system, and how much is due to the

process of discretizing the system.

193

Chapter 5

Generalized Variational Integrators

Abstract

In this chapter, we introduce generalized Galerkin variational integrators, which are a

natural generalization of discrete variational mechanics, whereby the discrete action,

as opposed to the discrete Lagrangian, is the fundamental object. This is achieved

by approximating the action integral with appropriate choices of a finite-dimensional

function space that approximate sections of the configuration bundle and numeri-

cal quadrature to approximate the integral. We discuss how this general framework

allows us to recover higher-order Galerkin variational integrators, asynchronous vari-

ational integrators, and symplectic-energy-momentum integrators. In addition, we

will consider function spaces that are not parameterized by field values evaluated at

nodal points, which allows the construction of Lie group, multiscale, and pseudospec-

tral variational integrators. The construction of pseudospectral variational integrators

is illustrated by applying it to the (linear) Schrodinger equation. G-invariant discrete

Lagrangians are constructed in the context of Lie group methods through the use of

natural charts and interpolation at the level of the Lie algebra. The reduction of these

G-invariant Lagrangians yield a higher-order analogue of discrete Euler–Poincare re-

duction. By considering nonlinear approximation spaces, spatio-temporally adaptive

variational integrators can be introduced as well.

5.1 Introduction

We will review some of the previous work on discrete mechanics and their multisymplectic gener-

alizations (see, for example, Marsden et al. [1998, 2001]), before introducing a general formulation

194

of discrete mechanics that recovers higher-order variational integrators (see, for example, Marsden

and West [2001]), asynchronous variational integrators (see, for example, Lew et al. [2003]), as well

symplectic-energy-momentum integrators (see, for example, Kane et al. [1999]).

While discrete variational integrators exhibit desirable properties such as symplecticity, momen-

tum preservation, and good energy behavior, it does not address other important issues in numerical

analysis, such as adaptivity and approximability. Generalized variational integrators are introduced

with a view towards addressing such issues in the context of discrete variational mechanics.

By formulating the construction of a generalized variational integrator in terms of the choice of

a finite-dimensional function space and a numerical quadrature scheme, we are able to draw upon

the extensive literature on approximation theory and numerical quadrature to construct variational

schemes that are appropriate for a larger class of problems. Within this framework, we will introduce

multiscale, spatio-temporally adaptive, Lie group, and pseudospectral variational integrators.

5.1.1 Standard Formulation of Discrete Mechanics

The standard formulation of discrete variational mechanics (see, for example, Marsden and West

[2001]) is to consider the discrete Hamilton’s principle,

δSd = 0,

where the discrete action sum, Sd : Qn+1 → R, is given by

Sd(q0, q1, . . . , qn) =n−1∑i=0

Ld(qi, qi+1).

The discrete Lagrangian, Ld : Q×Q→ R, is a generating function of the symplectic flow, and is

an approximation to the exact discrete Lagrangian,

Lexactd (q0, q1) =

∫ h

0

L(q01(t), q01(t))dt,

where q01(0) = q0, q01(h) = q1, and q01 satisfies the Euler–Lagrange equation in the time interval

(0, h). The exact discrete Lagrangian is related to the Jacobi solution of the Hamilton–Jacobi

equation. The discrete variational principle then yields the discrete Euler–Lagrange (DEL)

equation,

D2Ld(q0, q1) +D1Ld(q1, q2) = 0,

which yields an implicit update map (q0, q1) 7→ (q1, q2) that is valid for initial conditions (q0, q1)

that as sufficiently close to the diagonal of Q×Q.

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5.1.2 Multisymplectic Geometry

The generalization of the variational principle to the setting of partial differential equations involves

a multisymplectic formulation (see, for example, Marsden et al. [1998, 2001]). Here, the base space

X consists of the independent variables, which are denoted by (x0, . . . , xn), where x0 is time, and

x1, . . . , xn are space variables.

The independent or field variables, denoted (q1, . . . , qm), form a fiber over each space-time base-

point. The set of independent variables, together with the field variables over them, form a fiber

bundle, π : Y → X , referred to as the configuration bundle. The configuration of the system is

specified by giving the field values at each space-time point. More precisely, this can be represented

as a section of Y over X , which is a continuous map q : X → Y , such that π q = 1X . This means

that for every x ∈ X , q(x) is in the fiber over x, which is π−1(x).

In the case of ordinary differential equations, the Lagrangian is dependent on the position vari-

able, and its time derivative, and the action integral is obtained by integrating the Lagrangian in

time. In the multisymplectic case, the Lagrangian density is dependent on the field variables, and

the derivatives of the field variables with respect to the space-time variables, and the action integral

is obtained by integrating the Lagrangian density over a region of space-time.

The analogue of the tangent bundle TQ in the multisymplectic setting is referred to as the first

jet bundle J1Y , which consists of the configuration bundle Y , together with the first derivatives of

the field variables with respect to independent variables. We denote these as,

vij = qi,j =∂qi

∂xj,

for i = 1, . . . ,m, and j = 0, . . . , n.

We can think of J1Y as a fiber bundle over X . Given a section q : X → Y , we obtain its first

jet extension, j1q : X → J1Y , that is given by

j1q(x0, . . . , xn) =(x0, . . . , xn, q1, . . . , qm, q1,1, . . . , q

m,n

),

which is a section of the fiber bundle J1Y over X .

The Lagrangian density is a map L : J1Y → Ωn+1(X ), and the action integral is given by

S(q) =∫XL(j1q),

and then Hamilton’s principle states that

δS = 0.

196

We will see in the next subsection how this allows us to construct multisymplectic variational inte-

grators.

5.1.3 Multisymplectic Variational Integrator

We introduce a multisymplectic variational integrator through the use of a simple but illustrative

example. We consider a tensor product discretization of (1 + 1)-space-time, given by

•

•

•

•

•

•

•

•

•

xi−1 xi xi+1

tj−1

tj

tj+1

∆t

∆x

and tensor product shape functions given by

ϕi,j(x, t) =

xi xi+1

1

//

OO

????

????

?

⊗

tj tj+1

1

//

OO

????

????

?

We construct the discrete Lagrangian as follows,

Ld(qi,j , qi+1,j , qi,j+1, qi+1,j+1) =∫

[xi,xi+1]

∫[tj ,tj+1]

L(j1(∑i+1

a=i

∑j+1

b=jqa,bϕa,b

)),

where qi,j ' q(i∆x, j∆t).

Consider a variation that varies the value of qi,j , and leaves the other degrees of freedom fixed,

then we obtain the following discrete Euler–Lagrange equation,

D1Ld(qi,j , qi+1,j , qi,j+1, qi+1,j+1)+D2Ld(qi−1,j , qi,j , qi−1,j+1, qi,j+1)

+D3Ld(qi,j−1, qi+1,j−1, qi,j , qi+1,j)

+D4Ld(qi−1,j−1, qi,j−1, qi−1,j , qi,j) = 0 ,

197

In general, when we take variations in a degree of freedom, we will have a discrete Euler–Lagrange

equation that involves all terms in the discrete action sum that are associated with regions in space-

time that overlap with the support of the shape function associated with that degree of freedom.

5.2 Generalized Galerkin Variational Integrators

There are a few essential observations that go into constructing a general framework that en-

compasses the prior work on variational integrators, asynchronous variational integrators, and

symplectic-energy-momentum integrators, while yielding generalizations that allow the construction

of multiscale, spatio-temporally adaptive, Lie group, and pseudospectral variational integrators.

The first is that a generalized Galerkin variational integrator involves the choice of a finite-

dimensional function space that discretizes a section of the configuration bundle, and the second is

that we approximate the action integral though a numerical quadrature scheme to yield a discrete

action sum. The discrete variational equations we obtain from this discrete variational principle are

simply the Karush–Kuhn–Tucker (KKT) conditions (see, for example, Nocedal and Wright [1999])

with respect to the degrees of freedom that generate the finite-dimensional function space.

To recap, the choices which are made in discretizing a variational problem are:

1. A finite-dimensional function space to represent sections of the configuration bundle.

2. A numerical quadrature scheme to evaluate the action integral.

Given these two choices, we obtain an expression for the discrete action in terms of the degrees of

freedom, and the KKT conditions for the discrete action to be stationary with respect to variations

in the degrees of freedom yield the generalized discrete Euler–Lagrange equations.

Current variational integrators are based on piecewise polynomial interpolation, with function

spaces that are parameterized by the value of the field variables at nodal points, and a set of

internal points. By relaxing the condition that the interpolation is piecewise, we will be able to

consider pseudospectral discretizations, and by relaxing the condition that the parameterization is

in terms of field values, we will be able to consider Lie group variational integrators. By considering

shape functions motivated by multiscale finite elements (see, for example, Hou and Wu [1999];

Efendiev et al. [2000]; Chen and Hou [2003]), we will obtain multiscale variational integrators. And

by generalizing the approach used in symplectic-energy-momentum integrators, and considering

nonlinear approximation spaces (see, for example, DeVore [1998]), we will be able to introduce

spatio-temporally adaptive variational integrators.

As we will see, there is nothing canonical about the form of the discrete Euler–Lagrange equations,

or the notion that the discrete Lagrangian is a map Ld : Q×Q→ R. These expressions arise because

198

of the finite-dimensional function space which has been chosen.

5.2.1 Special Cases of Generalized Galerkin Variational Integrators

In this subsection, we will show how higher-order Galerkin variational integrators, multisymplectic

variational integrators, and symplectic-energy-momentum integrators are all special cases of gener-

alized Galerkin variational integrators.

Higher-Order Galerkin Variational Integrators. In the case of higher-order Galerkin vari-

ational integrators, we have chosen a piecewise interpolation for each time interval [0, h], that is

parameterized by control points q10 , . . . qs0, corresponding to the value of the curve at a set of control

times 0 = d0 < d1 < . . . < ds−1 < ds = 1. The interpolation within each interval [0, h] is given by

the unique degree s polynomial qd(t; qν0 , h), such that qd(dνh) = gν0 , for ν = 0, . . . , s.

Q

t0 d1h d2h ds−2h ds−1h h

•

•

•q10

..... ...............•

••qs−10

q00q20

qs−20

qs0

//

OO

Figure 5.1: Polynomial interpolation used in higher-order Galerkin variational integrators.

By an appropriate choice of quadrature scheme, we can break up the action integral into pieces,

which we denote by

Sid(qνi ) ≈∫ h

0

L(qd(t; qνi , h), qd(t; qνi , h))dt.

If we further require that the piecewise defined curve is continuous at the node points, we obtain

the following augmented discrete action,

Sd(qνi i=0,...,N−1

ν=0,...,s

)=N−1∑i=0

Sid (qνi sν=0)−N−2∑i=0

λi(qsi − q0i+1).

The discrete action is stationary when

∂Sid∂qsi

(qνi ) = λi,

199

∂Sid∂q0i+1

(qνi+1) = −λi,

∂Sid∂qji

(qνi ) = 0.

We can identify the pieces of the discrete action with the discrete Lagrangian, Ld : Q×Q→ R, by

setting

Ld(qi, qi+1) = Sid(qνi ), (5.2.1)

where, q0i = qi, qsi = qi+1, and the other qji ’s are defined implicitly by the system of equations

∂Sid∂qji

(qνi ) = 0, (5.2.2)

for ν = 1, . . . , s− 1. Once we have made this identification, we have that

−D1Ld(qi+1, qi+2) = −Si+1d

∂q0i+1

(qνi+1)

= λi

=∂Sid∂qsi

(qνi )

= D2Ld(qi, qi+1),

from which we recover the discrete Euler–Lagrange equation,

D1Ld(qi+1, qi+2) +D2Ld(qi, qi+1) = 0.

The DEL equations, together with the definition of the discrete Lagrangian, given in Equations 5.2.1

and 5.2.2, yield a higher-order Galerkin variational integrator. As we have shown, it is only because

the interpolation was piecewise that we were able to decompose the equations into a DEL equation,

and a set of equations that define the discrete Lagrangian in terms of conditions on the internal

control points.

Multisymplectic Variational Integrators. Recall the example of a multisymplectic variational

integrator we introduced in §5.1.3, where we used tensor product linear shape functions with localized

supports to discretize the configuration bundle. The discrete action is then given by

Sd(qa,ba=0,...,M−1

b=0,...,N−1

)=∫

[x0,xM ]

∫[t0,tN ]

L(j1( N∑a=0

N∑b=0

qa,bϕa,b

))

200

=M−1∑i=0

N−1∑j=0

∫[xi,xi+1]

∫[tj ,tj+1]

L(j1( N∑a=0

N∑b=0

qa,bϕa,b

))

=M−1∑i=0

N−1∑j=0

∫[xi,xi+1]

∫[tj ,tj+1]

L(j1( i+1∑a=i

j+1∑b=j

qa,bϕa,b

)),

where we first decomposed the integral into pieces, and then used the local support of the shape

functions to simplify the inner sums over qa,bϕa,b. As before, it is due to the local support of the

shape functions that we can express the discrete action as

Sd =M−1∑i=0

N−1∑j=0

Ld(qi,j , qi+1,j , qi,j+1, qi+1,j+1),

where

Ld(qi,j , qi+1,j , qi,j+1, qi+1,j+1) =∫

[xi,xi+1]

∫[tj ,tj+1]

L(j1(∑i+1

a=i

∑j+1

b=jqa,bϕa,b

)),

This localized support is also the reason why the discrete Euler–Lagrange equation in this case

consists of four terms, since to each degree of freedom, there are four other degrees of freedom that

have shape functions with overlapping support. In the case of ordinary differential equations, this

was two. In general, for tensor product meshes of (n + 1)-space-times, with tent function shape

functions, the number of terms in the discrete Euler–Lagrange equation will be 2n+1. In contrast,

for pseudospectral variational integrators with m spatial degrees of freedom per time level, and k

degrees of freedom per piecewise polynomial in time, each of the m(k − 1) discrete Euler–Lagrange

equations will involve m(k − 1) terms.

This is simply a reflection of the fact that shape functions with compact support yield schemes

with banded matrix structure, whereas pseudospectral and spectral methods tend to yield fuller

matrices. The payoff in using pseudospectral and spectral methods for problems with smooth or an-

alytic solutions is due to the approximation theoretic property that these solutions are approximated

at an exponential rate of accuracy by spectral expansions.

Symplectic-Energy-Momentum Integrators. In the case of symplectic-energy-momentum in-

tegrators, the degrees of freedom involve both the base variables and the field variables. We will

first derive a second-order symplectic-energy momentum integrator, and in the next section, we will

derive a higher-order generalization. We choose a piecewise linear interpolation for our configura-

tion bundle. Each piece is parameterized by the endpoint values of the field variable q0i , q1i , and the

endpoint times h0i , h

1i , and we approximate the action integral using the midpoint rule. To ensure

continuity, we require that, q1i = q0i+1, and h1i = h0

i+1.

201

Remark 5.1. The approach of allowing each piecewise defined curve to float around freely, and

imposing the continuity conditions using Lagrange multipliers was used in Lall and West [2003]

to unify the formulation of discrete variational mechanics and optimal control. Through the use

of a primal-dual formalism, discrete analogues of Hamiltonian mechanics and the Hamilton–Jacobi

equation were also introduced.

This yields the following discrete action,

Sd =N−1∑i=0

(h1i − h0

i )L(q0i + q1i

2,q1i − q0ih1i − h0

1

)−N−2∑i=0

λi(q1i − q0i+1)−N−2∑i=0

ωi(h1i − h0

i+1).

To simplify the expressions, we define

hi ≡ h1i − h0

i ,

qi+ 12≡ q0i + q1i

2,

qi+ 12≡ q1i − q0i

hi.

Then, the variational equations are given by

0 =L(qi+ 1

2, qi+ 1

2

)− hi

∂L

∂q

(qi+ 1

2, qi+ 1

2

) 1hiqi+ 1

2− ωi, for i = 0, . . . , N − 2,

0 =− L(qi+ 1

2, qi+ 1

2

)+ hi

∂L

∂q

(qi+ 1

2, qi+ 1

2

) 1hiqi+ 1

2+ ωi−1, for i = 1, . . . , N − 1,

0 =hi

[∂L

∂q

(qi+ 1

2, qi+ 1

2

) 12

+∂L

∂q

(qi+ 1

2, qi+ 1

2

) 1hi

]− λi, for i = 0, . . . , N − 2,

0 =hi

[∂L

∂q

(qi+ 1

2, qi+ 1

2

) 12− ∂L

∂q

(qi+ 1

2, qi+ 1

2

) 1hi

]+ λi−1, for i = 1, . . . , N − 1,

q1i =q0i+1, for i = 0, . . . , N − 2,

h1i =h0

i+1, for i = 0, . . . , N − 2.

If we define the discrete Lagrangian to be

Ld(qi, qi+1, hi) ≡ hiL

(qi + qi+1

2,qi+1 − qi

hi

),

and the discrete energy to be

Ed(qi, qi+1, hi) ≡ −D3Ld(qi, qi+1, hi)

= −L(qi + qi+1

2,qi+1 − qi

hi

)+ hi

∂L

∂q

(qi + qi+1

2,qi+1 − qi

hi

)1hi

qi+1 − qihi

,

202

and identify points as follows,

qi = q1i−1 = q0i ,

hi = h1i−1 = h0

i ,

we obtain

0 =Ed(qi, qi+1, hi)− ωi, for i = 0, . . . , N − 2,

0 =− Ed(qi, qi+1, hi) + ωi−1, for i = 1, . . . , N − 1,

0 =D2Ld(qi, qi+1, hi)− λi, for i = 0, . . . , N − 2,

0 =D1Ld(qi, qi+1, hi) + λi−1, for i = 1, . . . , N − 1.

After eliminating the Lagrange multipliers, we obtain the conservation of discrete energy equation,

Ed(qi, qi+1, hi) = Ed(qi+1, qi+2, hi+1),

and the discrete Euler–Lagrange equation,

D2Ld(qi, qi+1, hi) +D1Ld(qi+1, qi+2, hi+1) = 0.

This recovers the results obtained in Kane et al. [1999], but the derivation is new. In §5.5, we will

see how this derivation can be generalized to yield a higher-order scheme.

Discrete Action as the Fundamental Object. The message of this section is that the discrete

action is the fundamental object in discrete mechanics, as opposed to the discrete Lagrangian. In

instances whereby the shape function associated with individual degrees of freedom have localized

supports, it is possible to decompose the discrete action into terms that can be identified with

discrete Lagrangians. While this approach might seem artificial at first, we will find that in discussing

pseudospectral variational integrators, it does not make sense to break up the discrete action into

individual pieces.

5.3 Lie Group Variational Integrators

In this section, we will introduce higher-order Lie group variational integrators. The basic idea

behind all Lie group techniques is to express the update map of the numerical scheme in terms of

203

the exponential map,

g1 = g0 exp(ξ01) ,

and thereby reduce the problem to finding an appropriate Lie algebra element ξ01 ∈ g, such that the

update scheme has the desired order of accuracy. This is a desirable reduction, as the Lie algebra is

a vector space, and as such the interpolation of elements can be easily defined. In our construction,

the interpolatory method we use on the Lie group relies on interpolation at the level of the Lie

algebra.

For a more in depth review of Lie group methods, please refer to Iserles et al. [2000]. In the case

of variational Lie group methods, we will express the variational problem in terms of finding Lie

algebra elements, such that the discrete action is stationary.

As we will consider the reduction of these higher-order Lie group integrators in the next section,

we will chose a construction that yields a G-invariant discrete Lagrangian whenever the continuous

Lagrangian is G-invariant. This is achieved through the use of G-equivariant interpolatory functions,

and in particular, natural charts on G.

5.3.1 Galerkin Variational Integrators

We first recall the construction of higher-order Galerkin variational integrators, as originally de-

scribed in Marsden and West [2001]. Given a Lie group G, the associated state space is given by

the tangent bundle TG. In addition, the dynamics on G is described by a Lagrangian, L : TG→ R.

Given a time interval [0, h], the path space is defined to be

C(G) = C([0, h], G) = g : [0, h] → G | g is a C2 curve,

and the action map, S : C(G) → R, is given by

S(g) ≡∫ h

0

L(g(t), g(t))dt.

We approximate the action map, by numerical quadrature, to yield Ss : C([0, h], G) → R,

Ss(g) ≡ hs∑i=1

biL(g(cih), g(cih)),

where ci ∈ [0, 1], i = 1, . . . , s are the quadrature points, and bi are the quadrature weights.

Recall that the discrete Lagrangian should be an approximation of the form

Ld(g0, g1, h) ≈ extg∈C([0,h],G),g(0)=g0,g(h)=g1

S(g) .

204

If we restrict the extremization procedure to the subspace spanned by the interpolatory function

that is parameterized by s + 1 internal points, ϕ : Gs+1 → C([0, h], G), we obtain the following

discrete Lagrangian,

Ld(g0, g1) = extgν∈G;g0=g0;gs=g1

S(Tϕ(gν ; ·))

= extgν∈G;g0=g0;gs=g1

hs∑i=1

biL(Tϕ(gν ; cih)).

The interpolatory function is G-equivariant if

ϕ(ggν ; t) = gϕ(gν ; t).

Lemma 5.1. If the interpolatory function ϕ(gν ; t) is G-equivariant, and the Lagrangian, L : TG→

R, is G-invariant, then the discrete Lagrangian, Ld : G×G→ R, given by

Ld(g0, g1) = extgν∈G;g0=g0;gs=g1

hs∑i=1

biL(Tϕ(gν ; cih)),

is G-invariant.

Proof.

Ld(gg0, gg1) = extgν∈G;g0=gg0;gs=gg1

hs∑i=1


= extgν∈g−1G;g0=g0;gs=g1

hs∑i=1

biL(Tϕ(ggν ; cih)),


h

s∑i=1

biL(TLg · Tϕ(gν ; cih)),


hs∑i=1


= Ld(g0, g1),

where we used the G-equivariance of the interpolatory function in the third equality, and the G-

invariance of the Lagrangian in the forth equality.

Remark 5.2. While G-equivariant interpolatory functions provide a computationally efficient method

of constructing G-invariant discrete Lagrangians, we can construct a G-invariant discrete Lagrangian

(when G is compact) by averaging an arbitrary discrete Lagrangian. In particular, given a discrete

205

Lagrangian Ld : Q×Q→ R, the averaged discrete Lagrangian, given by

Ld(q0, q1) =1|G|

∫g∈G

Ld(gq0, gq1)dg

is G-equivariant. Therefore, in the case of compact symmetry groups, a G-invariant discrete La-

grangian always exists.

5.3.2 Natural Charts

Following the construction in Marsden et al. [1999], we use the group exponential map at the identity,

expe : g → G, to construct a G-equivariant interpolatory function, and a higher-order discrete

Lagrangian. As shown in Lemma 5.1, this construction yields a G-invariant discrete Lagrangian if

the Lagrangian itself is G-invariant.

In a finite-dimensional Lie group G, expe is a local diffeomorphism, and thus there is an open

neighborhood U ⊂ G of e such that exp−1e : U → u ⊂ g. When the group acts on the left, we obtain

a chart ψg : LgU → u at g ∈ G by

ψg = exp−1e Lg−1 .

Lemma 5.2. The interpolatory function given by

ϕ(gν ; τh) = ψ−1g0

(∑s

ν=0ψg0(gν)lν,s(τ)

),

is G-equivariant.

Proof.

ϕ(ggν ; τh) = ψ−1(gg0)

(∑s

ν=0ψgg0(ggν)lν,s(τ)

)= Lgg0 expe

(∑s

ν=0exp−1

e ((gg0)−1(ggν))lν,s(τ))

= LgLg0 expe(∑s

ν=0exp−1

e ((g0)−1g−1ggν)lν,s(τ))

= Lgψ−1g0

(∑s

ν=0exp−1

e L(g0)−1(gν)lν,s(τ))

= Lgψ−1g0

(∑s

ν=0ψg0(gν)lν,s(τ)

)= Lgϕ(gν ; τh).

Remark 5.3. In the proof that ϕ is G-equivariant, it was important that the base point for the chart

should transform in the same way as the internal points gν . As such, the interpolatory function will

be G-equivariant for a chart that it based at any one of the internal points gν that parameterize

the function, but will not be G-equivariant if the chart is based at a fixed g ∈ G. Without loss of

206

generality, we will consider the case when the chart is based at the first point g0.

We will now consider a discrete Lagrangian based on the use of interpolation in a natural chart,

which is given by

Ld(g0, g1) = extgν∈G;g0=g0;gs=g−1

0 g1

hs∑i=1

biL(Tϕ(gνsν=0; cih) .

To further simplify the expression, we will express the extremal in terms of the Lie algebra elements

ξν associated with the ν-th control point. This relation is given by

ξν = ψg0(gν) ,

and the interpolated curve in the algebra is given by

ξ(ξν ; τh) =s∑

κ=0

ξκ lκ,s(τ),

which is related to the curve in the group,

g(gν ; τh) = g0 exp(ξ(ψg0(gν); τh)).

The velocity ξ = g−1g is given by

ξ(τh) = g−1g(τh) =1h

s∑κ=0

ξκ˙lκ,s(τ).

Using the standard formula for the derivative of the exponential,

Tξ exp = TeLexp(ξ) · dexpadξ,

where

dexpw =∞∑n=0

wn

(n+ 1)!,

we obtain the following expression for discrete Lagrangian,

Ld(g0, g1) = extξν∈g;ξ0=0;ξs=ψg0 (g1)

hs∑i=1

biL(Lg0 exp(ξ(cih)),

Texp(ξ(cih))Lg0 · TeLexp(ξ(cih)) · dexpadξ(cih)(ξ(cih))

).

More explicitly, we can compute the conditions on the Lie algebra elements for the expression above

207

to be extremal. This implies that

Ld(g0, g1) = hs∑i=1

biL(Lg0 exp(ξ(cih)), Texp(ξ(cih))Lg0 · TeLexp(ξ(cih)) · dexpadξ(cih)

(ξ(cih)))

with ξ0 = 0, ξs = ψg0(g1), and the other Lie algebra elements implicitly defined by

0 = hs∑i=1

bi

[∂L

∂g(cih)Texp(ξ(cih))Lg0 · TeLexp(ξ(cih)) · dexpadξ(cih)

lν,s(ci)

+1h

∂L

∂g(cih)T 2

exp(ξ(cih))Lexp(ξ(cih)) · T

2e Lexp(ξ(cih)) · ddexpadξ(cih)

˙lν,s(ci)

],

for ν = 1, . . . , s− 1, and where

ddexpw =∞∑n=0

wn

(n+ 2)!.

This expression for the higher-order discrete Lagrangian, together with the discrete Euler–Lagrange

equation,

D2Ld(g0, g1) +D1Ld(g1, g2) = 0 ,

yields a higher-order Lie group variational integrator.

5.4 Higher-Order Discrete Euler–Poincare Equations

In this section, we will apply discrete Euler–Poincare reduction (see, for example, Marsden et al.

[1999]) to the Lie group variational integrator we derived previously, to construct a higher-order

generalization of discrete Euler–Poincare reduction.

5.4.1 Reduced Discrete Lagrangian

We first proceed by computing an expression for the reduced discrete Lagrangian in the case when

the Lagrangian is G-invariant. Recall that our discrete Lagrangian uses G-equivariant interpolation,

which, when combined with the G-invariance of the Lagrangian, implies that the discrete Lagrangian

is G-invariant as well. We compute the reduced discrete Lagrangian,

ld(g−10 g1) ≡ Ld(g0, g1)

= Ld(e, g−10 g1)

= extξν∈g;ξ0=0;ξs=log(g−1

0 g1)h

s∑i=1

biL(Le exp(ξ(cih)),

Texp(ξ(cih))Le · TeLexp(ξ(cih)) · dexpadξ(cih)(ξ(cih))

)

208

= extξν∈g;ξ0=0;ξs=log(g−1

0 g1)h

s∑i=1

biL(

exp(ξ(cih)), TeLexp(ξ(cih)) · dexpadξ(cih)(ξ(cih))

).

Setting ξ0 = 0, and ξs = log(g−10 g1), we can solve the stationarity conditions for the other Lie

algebra elements ξνs−1ν=1 using the following implicit system of equations,

0 = hs∑i=1

bi

[∂L

∂g(cih)TeLexp(ξ(cih)) · dexpadξ(cih)

lν,s(ci)

+1h

∂L

∂g(cih)T 2

e Lexp(ξ(cih)) · ddexpadξ(cih)

˙lν,s(ci)

]

where ν = 1, . . . , s− 1.

This expression for the reduced discrete Lagrangian is not fully satisfactory however, since it

involves the Lagrangian, as opposed to the reduced Lagrangian. If we revisit the expression for the

reduced discrete Lagrangian,

ld(g−10 g1) = ext

ξν∈g;ξ0=0;ξs=log(g−10 g1)

h

s∑i=1

biL(


),

we find that by G-invariance of the Lagrangian, each of the terms in the summation,

L(


),

can be replaced by

l(

dexpadξ(cih)(ξ(cih))

),

where l : g → R is the reduced Lagrangian given by

l(η) = L(Lg−1g, TLg−1 g) = L(e, η),

where η = TLg−1 g ∈ g.

From this observation, we have an expression for the reduced discrete Lagrangian in terms of the

reduced Lagrangian,

ld(g−10 g1) = ext

ξν∈g;ξ0=0;ξs=log(g−10 g1)

hs∑i=1

bil(


).

As before, we set ξ0 = 0, and ξs = log(g−10 g1), and solve the stationarity conditions for the other

209

Lie algebra elements ξνs−1ν=1 using the following implicit system of equations,

0 = hs∑i=1

bi

[∂l

∂η(cih) ddexpadξ(cih)

˙lν,s(ci)

],

where ν = 1, . . . , s− 1.

5.4.2 Discrete Euler–Poincare Equations

As shown above, we have constructed a higher-order reduced discrete Lagrangian that depends on

fkk+1 ≡ gkg−1k+1.

We will now recall the derivation of the discrete Euler–Poincare equations, introduced in Marsden

et al. [1999]. The variations in fkk+1 induced by variations in gk, gk+1 are computed as follows,

δfkk+1 = −g−1k δgkgk−1gk+1 + g−1

k δgk+1

= TRfkk+1(−g−1k δgk + Adfkk+1 gk+1δgk+1) .

Then, the variation in the discrete action sum is given by

δS =N−1∑k=0

l′d(fkk+1)δfkk+1

=N−1∑k=0

l′d(fkk+1)TRfkk+1(−g−1k δgk + Adfkk+1 gk+1δgk+1)

=N−1∑k=1

[l′d(fk−1k)TRfk−1k

Adfk−1k−l′d(fkk+1)TRfkk+1

]ϑk ,

with variations of the form ϑk = g−1k δgk. In computing the variation of the discrete action sum, we

have collected terms involving the same variations, and used the fact that ϑ0 = ϑN = 0. This yields

the discrete Euler–Poincare equation,

l′d(fk−1k)TRfk−1kAdfk−1k

−l′d(fkk+1)TRfkk+1 = 0, k = 1, . . . , N − 1.

For ease of reference, we will recall the expressions from the previous subsection that define the

higher-order reduced discrete Lagrangian,

ld(fkk+1) = hs∑i=1

bil(


),

210

where

ξ(ξν ; τh) =s∑

κ=0

ξκ lκ,s(τ) ,

and

ξ0 = 0 ,

ξs = log(fkk+1) ,

and the remaining Lie algebra elements ξνs−1ν=1, are defined implicitly by

0 = hs∑i=1

bi

[∂l

∂η(cih) ddexpadξ(cih)

˙lν,s(ci)

],

for ν = 1, . . . , s− 1, and where

ddexpw =∞∑n=0

wn

(n+ 2)!.

When the discrete Euler–Poincare equation is used in conjunction with the higher-order reduced

discrete Lagrangian, we obtain the higher-order Euler–Poincare equations.

5.5 Higher-Order Symplectic-Energy-Momentum Variational

Integrators

In this section, we will generalize our new derivation of the symplectic-energy-momentum preserving

variational integrators (see, Kane et al. [1999]) to yield integrators with higher-order accuracy.

As before, we consider a piecewise interpolation, with both the control points in the field variables,

qνi , and the endpoints of the interval, h0i , h

1i , as degrees of freedom. The continuity conditions for

this function space are qsi = q0i+1, and h1i = h0

i+1. Then, we have that the discrete action is given by

Sd =N−1∑i=0

(h1i − h0

i )s∑j=1

bjL(qi(cj(h1i − h0

i ); qνi ), qi(cj(h

1i − h0

i ); qνi ))

−N−2∑i=0

λi(qsi − q0i+1)−N−2∑i=0

ωi(h1i − h0

i+1),

where

qi(τ(h1i − h0

i ); qνi ) =

s∑κ=0

qκi lκ,s(τ),

211

qi(τ(h1i − h0

i ); qνi ) =

1h1i − h0

i

s∑κ=0

qκi˙lκ,s(τ).

To simplify the expressions, we define hi ≡ h1i − h0

i . Then, the variational equations are given by

0 =s∑j=1

bjL(qi(cjhi), qi(cjhi))− hi

s∑j=1

bj∂L

∂q(cjhi)

1hiqi(cjhi)− ωi, for i = 0, . . . , N − 2,

0 =−s∑j=1

bjL(qi(cjhi), qi(cjhi)) + hi

s∑j=1

bj∂L

∂q(cjhi)

1hiqi(cjhi) + ωi−1, for i = 1, . . . , N − 1,

0 =his∑j=1

bj

[∂L

∂q(cjhi)ls,s(cj) +

1hi

∂L

∂q(cjhi)

˙ls,s(cj)

]− λi, for i = 0, . . . , N − 2,

0 =his∑j=1

bj

[∂L

∂q(cjhi)l0,s(cj) +

1hi

∂L

∂q(cjhi)

˙l0,s(cj)

]+ λi−1, for i = 1, . . . , N − 1,

0 =his∑j=1

bj

[∂L

∂q(cjhi)lν,s(cj) +

1hi

∂L

∂q(cjhi)

˙lν,s(cj)

],

for i = 0, . . . , N − 1,ν = 1, . . . , s− 1,

qsi =q0i+1, for i = 0, . . . , N − 2,

h1i =h0

i+1, for i = 0, . . . , N − 2.

We can eliminate the Lagrange multipliers, to yield

0 =s∑j=1

bjL(qi(cjhi), qi(cjhi))−s∑j=1

bj∂L

∂q(cjhi)qi(cjhi)

+s∑j=1

bjL(qi−1(cjhi−1), qi−1(cjhi−1))

−s∑j=1

bj∂L

∂q(cjhi−1)qi−1(cjhi−1), for i = 1, . . . , N − 1,

0 =his∑j=1

bj

[∂L

∂q(cjhi)ls,s(cj) +

1hi

∂L

∂q(cjhi)

˙ls,s(cj)

]

+ hi−1

s∑j=1

bj

[∂L

∂q(cjhi−1)l0,s(cj) +

1hi−1

∂L

∂q(cjhi−1)

˙l0,s(cj)

], for i = 1, . . . , N − 1,

0 =his∑j=1

bj

[∂L


1hi

∂L

∂q(cjhi)

˙lν,s(cj)

],

for i = 0, . . . , N − 1,ν = 1, . . . , s− 1,

qsi =q0i+1, for i = 0, . . . , N − 2,

h1i =h0

i+1, for i = 0, . . . , N − 2.

212

If we define the discrete Lagrangian as follows,

Ld(qi, qi+1, hi) ≡ hi

s∑j=1

bjL(qi(cjhi), qi(cjhi)),

where

qi(τhi; qνi ) =s∑

κ=0

qκi lκ,s(τ),

qi(τhi; qνi ) =1hi

s∑κ=0

qκi˙lκ,s(τ),

and q0i = qi, qsi = q1, and the remaining terms were defined implicitly by

0 = hi

s∑j=1

bj

[∂L


1hi

∂L

∂q(cjhi)

˙lν,s(cj)

],

then the equations reduce to the following,

Ed(qi, qi+1, hi) = − ∂

∂hi[Ld(qi, qi+1, hi)],

Ed(qi, qi+1, hi) = Ed(qi+1, qi+2, hi+1),

0 = D2Ld(qi, qi+1, hi) +D1Ld(qi+1, qi+2, hi+1).

which is a higher-order symplectic-energy-momentum variational integrator.

Solvability of the Energy Equation. It should be noted that the discrete energy conservation

equation is not necessarily solvable, in general, particularly near stationary points. This issue is

discussed in Kane et al. [1999]; Lew et al. [2004], and can be addressed by reformulating the discrete

energy conservation equation as an optimization problem that chooses the time step by minimizing

the discrete energy error squared. Clearly, reformulating the discrete energy conservation equation

yields the desired behavior whenever the discrete energy conservation equation can be solved, while

allowing the computation to proceed when discrete energy conservation cannot be achieved, albeit

with a slight energy error in that case. This does not degrade performance significantly, since

instances in which discrete energy conservation cannot be achieved are rare.

5.6 Spatio-Temporally Adaptive Variational Integrators

As is the case with all inner approximation techniques in numerical analysis, the quality of the

numerical solution we obtain is dependent on the rate at which the sequence of finite-dimensional

213

function spaces approximates the actual solution as the number of degrees of freedom is increased.

For problems that exhibit shocks, nonlinear approximation spaces (see, for example, DeVore

[1998]), as opposed to linear approximation spaces, are clearly preferable. Adaptive techniques

have been developed in the context of finite elements under the name of r-adaptivity and moving

finite elements (see, for example, Baines [1995]), and has been developed in a variational context

for elasticity in Thoutireddy and Ortiz [2003]. The standard motivation in discrete mechanics to

introduce function spaces that have degrees of freedom associated with the base space is to achieve

energy or momentum conservation, as discussed in §5.5, or Kane et al. [1999]; Oliver et al. [2004].

However, if the solution to be approximated exhibits shocks, nonlinear approximation techniques

achieve better results for a given number of degrees of freedom.

In this section, we will sketch the use of regularizing transformations of the base space, to achieve

a computational representation of sections of the configuration bundle that will yield more accurate

numerical results.

Consider the situation when we are representing a characteristic function using piecewise spline

interpolation. We show in Figure 5.2, the difference between linear and nonlinear approximation of

the characteristic function.

(a) Using equispaced nodes (b) Using adaptive nodes

Figure 5.2: Linear and nonlinear approximation of a characteristic function.

When the derivatives of the solution vary substantially in a spatially distributed manner, we

obtain additional accuracy, for a fixed representation cost, if we allow nodal points to cluster near

regions of high curvature. It is therefore desirable to consider variational integrators based on

function spaces that are parameterized by both the position of the nodal points on the base space,

as well as the field values over the nodes.

This is represented by having a regular grid for the computational domain R, which is then

mapped to the physical base space X , as shown in Figure 5.3.

214

7→

Figure 5.3: Mapping of the base space from the computational to the physical domain.

The sections of the configuration bundle factor as follows,

Y

R ϕ//

q>>~~~~~~~X

q

OO

The mapping ϕ : R → X results in a regularized computational representation q : R → Y of the

original section q : X → Y . The relationship between the discrete section of the configuration bundle

and its computational representation is illustrated in Figure 5.4.

ϕ//

q

OO

q

OO

Figure 5.4: Factoring the discrete section.

The action integral is then given by

S(q) =∫XL(j1q) =

∫RL(j1q)|Dϕ|.

Thus, even though q : X → Y may exhibit shocks, the computational representation we work with,

q : R → Y , is substantially more regular, and consequently, a numerical quadrature scheme in R

215

applied to ∫RL(j1q)|Dϕ|

is significantly more accurate than the corresponding numerical quadrature scheme in X applied to

∫XL(j1q).

As such, spatio-temporally adaptive variational integrators achieve increased accuracy by allowing

the accurate representation of shock solutions using an adapted free knot representation, while using

a smooth computational representation to compute the action integral.

5.7 Multiscale Variational Integrators

In the work on multiscale finite elements (MsFEM), introduced and developed in Hou and Wu [1999];

Efendiev et al. [2000]; Chen and Hou [2003], shape functions that are solutions of the fast dynamics

in the absence of slow forces are constructed to yield finite element schemes that achieve convergence

rates that are independent of the ratio of fast to slow scales.

In constructing a multiscale variational integrator, we need to choose finite-dimensional function

spaces that do a good job of approximating the fast dynamics of the problem, when the slow variables

are frozen. In addition, we require an appropriate choice of numerical quadrature scheme to be able

to evaluate the action, which involves integrating a highly-oscillatory Lagrangian. In this section,

we will discuss how to go about making such choices of function spaces and quadrature methods.

We will start with a discussion of the multiscale finite element method, to illustrate the impor-

tance of a good choice of shape functions in computing solutions to problems with multiple scales.

After that, we will walk through the construction of a multiscale variational integrator for the case

of a planar pendulum with a stiff spring. Finally, we will discuss how we might proceed if we do not

possess knowledge of which variables, or forces, are fast or slow.

5.7.1 Multiscale Shape Functions

We will illustrate the idea of constructing shape functions that are solutions of the fast dynamics by

introducing a model multiscale second-order elliptic partial differential equation given by

∇ · a(x/ε)∇uε(x) = f(x),

216

with homogeneous boundary conditions. In the one-dimensional case, we can solve for the solution

analytically, and it has the form

uε(x) =∫ x

0

F (y)a(y/ε)

dy −

∫ 1

0F (y)a(y/ε)dy∫ 1

0dy

a(y/ε)

∫ x

0

dy

a(y/ε),

where F (x) =∫ x0f(y)dy. If we have nodal points at xiNi=0, then the appropriate multiscale shape

functions to adopt in this example is to use shape functions that are solutions of the homogeneous

problem at the element level. These shape functions ϕεi satisfy

∂∂x

(a(x/ε) ∂∂xϕ

εi

)= 0, for xi−1 < x < xi+1;

ϕεi(xi−1) = 0; ϕεi(xi+1) = 0; ϕεi(xi) = 1.

And they have the explicit form given by

ϕεi(x) =

[∫ xi

xi−1

dsa(s/ε)

]−1 [∫ xxi−1

dsa(s/ε)

], x ∈ [xi−1, xi] ;[∫ xi+1

xi

dsa(s/ε)

]−1 [∫ xi+1

xds

a(s/ε)

], x ∈ (xi, xi+1] ;

0, otherwise .

It can be shown that this will yield a numerical scheme that solves exactly for the solution at the

nodal points. This analysis is carried out in Appendix C.

As an example, we will compute the analytical solution for a(x) = 101+0.95 sin(2πx) , f(x) = x2, and

ε = 0.025. This is illustrated in Figure 5.5(a). What is particularly interesting is to compare the

zoomed plot of the exact solution and the multiscale shape function over the same interval, shown

in Figures 5.5(b) and 5.5(c), respectively. The multiscale finite element method is able to achieve

excellent results because the multiscale shape functions are able to capture the fast dynamics well.

In the next subsection, we will discuss how this insight is relevant in the construction of multiscale

variational integrators.

5.7.2 Multiscale Variational Integrator for the Planar Pendulum with a

Stiff Spring

As was shown previously, a shape function that captures the fast dynamics of a multiscale problem

is able to achieve superior accuracy when used for computation. While this idea has primarily been

used for problems with multiple spatial scales, it is natural to consider its application to a problem

with multiple temporal scales, such as the problem of the planar pendulum with a stiff spring, as


217

0 0.2 0.4 0.6 0.8 1−0.04

−0.03

−0.02

−0.01

0

0.01

(a) Exact solution

0 0.05 0.1 0.15 0.2 0.25−0.025

−0.02

−0.015

−0.01

−0.005

0

(b) Exact solution (zoomed)

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

(c) Multiscale shape function

Figure 5.5: Comparison of the multiscale shape function and the exact solution for the elliptic

problem.

218

Figure 5.6: Planar pendulum with a stiff spring

We will use this example to illustrate the issues that arise in constructing a multiscale variational

integrator. The variables are q = (a, θ), where a is the spring extension, and θ is the angle from the

vertical. The Lagrangian is given by

L(a, θ, a, θ) =m

2(x2 + y2)−mgy − k

2a2,

where

x = (l + a) sin θ,

y = −(l + a) cos θ,

x = a sin θ + θ(l + a) cos θ,

y = −a cos θ + θ(l + a) sin θ.

The Hamilton’s equations for the planar pendulum with a stiff spring are

a =pam,

θ =pθ

m(l + a)2,

pa = −ka+ gm cos θ +m(l + a)θ2,

pθ = −gm(l + a) sin θ.

The timescale arising from the mass-spring system is 2π√m/k. The timescale arising from the

planar pendulum system is 2π√l/g. The ratio of timescales is given by ε =

√mg/kl.

219

Multiscale Shape Function. In this problem, the fast scale is associated with the stiff spring,

and if we set the slow variable θ = 0, we obtain the equation

a =pam

= − k

ma,

which has solutions of the form

a(t) = a0 sin(√

k/m t)

+ a1 cos(√

k/m t).

We will now consider a well-resolved simulation of this system using the ode15s stiff solver from

Matlab, with parameters m = 1, g = 9.81, k = 10000, l = 1, giving a scale separation of ε = 0.0313.

The simulation results are show in Figure 5.7.

0 5 10 15 20−1

−0.5

0

0.5

1

(a) a(t)

0 5 10 15 20−2

−1

0

1

2

(b) θ(t)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

(c) a(t) (zoomed)

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

(d) 0.5 cos“p

k/m t”

Figure 5.7: Comparison of the multiscale shape function and the exact solution for the planar

pendulum with a stiff spring.

Clearly, if we wish to choose time steps that do not resolve the fast oscillations in a, but do

resolve the slow oscillations in θ, it would be desirable to include sin(√

k/m t), and cos

(√k/m t

)in the finite-dimensional function space used to interpolate a(t).

220

Evaluating the Discrete Lagrangian. Since we have chosen time steps that do not resolve the

fast oscillations, it follows that over the interval [0, h], the Lagrangian will oscillate rapidly as well.

In computing the discrete Lagrangian, it is therefore necessary to ensure that this highly-oscillatory

integral is well-approximated.

It is conventional wisdom, in numerical analysis, that the numerical quadrature of highly-

oscillatory integrals is a challenging problem requiring the use of many function evaluations. Re-

cently, there has been a series of papers, Iserles [2003a,b, 2004]; Iserles and Nørsett [2004], that pro-

vides an analysis of Filon-type quadrature schemes that provide an efficient and accurate method

of evaluating such integrals. The method is applicable to weighted integrals as well, but we will

summarize the results from §3 of Iserles [2003a] restricted to unweighted integrals, and refer the

reader to the original reference for an in-depth discussion and analysis.

The Filon-type method aims to evaluate an integral of the form

Ih[f ] =∫ h

0

f(x)eiωxdx = h

∫ h

0

f(hx)eiωxdx.

Given a set of distinct quadrature points, c1 < c2 < · · · < cν in [0, 1], the Filon-type quadrature

method is given by

QFh [f ] = hν∑i=1

bi(ihω)f(cih),

where

bi(ihω) =∫ 1

0

li(x)eihωxdx,

and li are the Legendre polynomials. Here, we draw attention to the fact that the quadrature weights

are dependent on hω.

If the quadrature points correspond to Gauss-Christoffel quadrature of order p, then the error

for the Filon-type method is given as follows.

O(hp+1), if hω 1

O(hν), if hω = O(1)

O(hν+1/(hω)), if hω 1

O(hν+1/(hω)2), if hω 1, c1 = 0, cν = 1.

Clearly, for highly-oscillatory functions, the last case, which corresponds to the Lobatto quadrature

points, is most desirable. Thus, it is appropriate to use the Filon-Lobatto method to evaluate the

discrete Lagrangian in our case.

221

Discrete Variational Equations. As we discussed previously, instead of looking for stationary

solutions to the discrete Hamilton’s principle in polynomial spaces, we will consider solutions that

are piecewise of the form

q(t; pj, ω, a0, a1) =(∑n

j=0pjt

j)

(1 + a0 sin(ωt) + a1 cos(ωt)) .

This function space approximates the highly-oscillatory nature of the solution well, in contrast

to a polynomial function space, thereby avoiding approximation-theoretic errors. The degrees of

freedom in this function space are pj, ω, a0, and a1. Since there are no distinguished degrees of

freedom that are responsible for the endpoint values of the curve, we need to impose continuity at

the nodes using a Lagrange multiplier. The augmented discrete action is given by

Sd =N−1∑i=0

∫ h

0

L(j1qi(t; pij, ωi, ai0, ai1))dt

+N−2∑i=0

λi(qi(h; pij, ωi, ai0, ai1)− qi+1(0; pi+1j , ωi+1, ai+1

0 , ai+11 )) ,

where each of the integrals are evaluated using the Filon-Lobatto method. Taking variations with

respect to the degrees of freedom yields an update map,

(pij, ωi, ai0, ai1

)7→(pi+1j , ωi+1, ai+1

0 , ai+11

),

which gives the multiscale variational integrator for the planar pendulum with a stiff spring.

5.7.3 Computational Aspects

Multiscale variational integrators have the advantage of directly accounting for the contribution of

the fast dynamics, thereby allowing the scheme to use significantly larger time-steps, while maintain-

ing accuracy and stability. It is possible to take advantage of knowledge about which of the variables,

or forces, are fast or slow, by using a low degree polynomial and oscillatory functions for the fast

variables, and a higher-order polynomial for the slow variables. In the absence of such information,

it is appropriate to use a function space with both polynomials and oscillatory functions, and apply

it to all the variables.

Recall that the Filon-type method has quadrature coefficients that depend on the frequency. As

such, the initial fast frequency has to be estimated numerically using a fully resolved computation

for a short period of time. Since both the function space and the quadrature weights depend on

the fast frequency ω, the resulting scheme is implicit and fairly nonlinear, and as such, it may be

expensive for large systems.

222

5.8 Pseudospectral Variational Integrators

The use of spectral expansions of the solution in space are particularly appropriate for highly accurate

simulations of the evolution of smooth solutions, such as those arising from quantum mechanics. We

will introduce pseudospectral variational integrators, and consider the Schrodinger equation as an

example.

In particular we will adopt the tensor product of a spectral expansion in space, and a polynomial

expansion in time. For example, we could have an interpolatory function of the form

ψ(x, (τ + l)∆t) =12π

N/2∑′

k=−N/2

eikx((1− τ)vlk + τ vl+1

k

),

which is the tensor product of a discrete Fourier expansion in space, and linear interpolation in time.

Here, the∑′ notation denotes a weighted sum where the terms with indices ±N/2 are weighted by

1/2, and the other terms are weighted by 1. See page 19 of Trefethen [2000] for a discussion of why

this is necessary to fix an issue with derivatives of the interpolant.

The degrees of freedom are given by vlk, which are the discrete Fourier coefficients. We will later

see how such an interpolation can be applied to the Schrodinger equation. The action integral can

be exactly evaluated for this class of shape functions, as we will see below.

It is straightforward to generalize the pseudospectral approach we present in this section to

a spectral variational integrator, with discrete Fourier expansions in space for periodic domains,

or Chebyshev expansions in space for non-periodic domains, and Chebyshev expansions in time.

This will however result in all the degrees of freedom on the space-time mesh being coupled, and

is therefore substantially more expensive computationally than the pseudospectral method. The

payoff for adopting the spectral approach is spectral accuracy, which is accuracy beyond all orders.

5.8.1 Variational Derivation of the Schrodinger Equation

Let H be a complex Hilbert space, for example, the space of complex-valued functions ψ on R3 with

the Hermitian inner product,

〈ψ1, ψ2〉 =∫ψ1(x)ψ2(x)d

3x ,

where the overbar denotes complex conjugation. We will present a Lagrangian derivation of the

Schrodinger equation, following worked example 9.1 on pages 568–569 of Jose and Saletan [1998].

Consider the Lagrangian density L given by

L(j1ψ) =i~2ψψ − ψψ − Hψψ,

223

where H : H → H is given by

Hψ = − ~2

2m∇2ψ + V ψ,

which yields

L(j1ψ) =i~2ψψ − ψψ − ~2

2m∇ψ · ∇ψ − V ψψ.

We take ψ,ψ as independent variables, and compute,

δ

∫Ldt =

∫ [(∂L∂ψ

δψ +∂L∂ψ

δψ +∂L∂∇ψ

δ∇ψ

)+

(∂L∂ψ

δψ +∂L∂ψ

δψ +∂L∂∇ψ

δ∇ψ

)]d3xdt

=∫ [(

∂L∂ψ

− ∂

∂t

∂L∂ψ

−∇ · ∂L∂∇ψ

)δψ +

(∂L∂ψ

− ∂

∂t

∂L∂ψ

−∇ · ∂L∂∇ψ

)δψ

]d3xdt

=∫ [(

i~2ψ − V ψ +

i~2ψ +

~2

2m∇2ψ

)δψ +

(i~2ψ − V ψ +

i~2ψ +

~2

2m∇2ψ

)δψ

]d3xdt,

where we integrated by parts, and neglected boundary terms as the variations vanish at the boundary

of the space-time region. Since the variations are arbitrary, we obtain the nonrelativistic (linear)

Schrodinger equation as a result,

i~ψ =− ~2

2m∇2 + V

ψ .

We note that the Lagrangian density is invariant under the internal phase shift given by

ψ 7→ eiεψ, ψ 7→ e−iεψ.

The space part of the multi-momentum map is given by

jk =∂L

∂(∂kψ)iψ +

∂L∂(∂kψ)

(−iψ) =i~2

2m(ψ∂kψ − ψ∂kψ),

and the time part is given by

j0 =∂L∂ψ

iψ − ∂L∂ψ

(−iψ) = −~2(ψψ − ψψ).

The norm of the wavefunction is automatically preserved by variational integrators, since the

norm is a quadratic invariant.

5.8.2 Pseudospectral Variational Integrator for the Schrodinger Equation

Consider a periodic domain [0, 2π], discretized with a discrete Fourier series expansion in space, and

a linear interpolation in time. Let N be an even integer, then, our computation is done on the

224

following mesh,

0 π 2π

x1 x2 xN/2 xN−1 xN• • • • • • • • • •

This implies that the grid spacing is given by

h =2πN.

The interpolation is given by

ψ(x, (τ + l)∆t) =12π

N/2∑′

k=−N/2


k

),

ψ(x, (τ + l)∆t) =1

2π∆t

N/2∑′

k=−N/2

eikx(vl+1k − vlk

),

ψ(x, (τ + l)∆t) =12π

N/2∑′

k=−N/2

e−ikx((1− τ)¯vlk + τ ¯vl+1

k

),

˙ψ(x, (τ + l)∆t) =1

2π∆t

N/2∑′

k=−N/2

e−ikx(¯vl+1k − ¯vlk

),

and the discrete Fourier transformation is given by

vj =12πh

N∑j=1

e−ikxjvj ,

for k = −N/2 + 1, . . . , N/2, and v−N/2 ≡ vN/2. Recall that

L(j1ψ) =i~2ψψ − ψψ − Hψψ,

where H : H → H is given by

Hψ = − ~2

2m∇2ψ + V ψ.

Furthermore, the potential V is expressed using a discrete Fourier expansion,

V (x) =12π

N/2∑′

k=−N/2

eikxVk .

225

In addition, we will need to introduce a normalization condition, so as to eliminate trivial solutions

of the partial differential equation. The normalization condition is

1 = 〈ψl, ψl〉 =∫ 2π

0

12π

N/2∑′

k=−N/2

eikxvlk

12π

N/2∑′

k=−N/2

e−ikx ¯vlk

dx =12π

N/2∑′′

k=−N/2

vlk¯vlk,

which is enforced using a Lagrange multiplier.

Discrete Action for the Schrodinger Equation. The discrete action in the space-time region

[0, 2π]× [l∆t, (l + 1)∆t] is given by

Sd =∫ (l+1)∆t

l∆t

∫ 2π

0

L(j1ψ)dxdt+ λl(1− 〈ψl, ψl〉)

=∫ (l+1)∆t

l∆t

∫ 2π

0

[i~2ψψ − ψ ˙ψ+

~2

2m∇2ψψ − V ψψ

]dxdt+ λl

1− 12π

N/2∑′′

k=−N/2

vlk¯vlk

=∫ 1

0

∫ 2π

0

i~2

12π∆t

N/2∑′

k=−N/2

eikx(vl+1k − vlk

) 12π

N/2∑′

k=−N/2


k

)−

12π

N/2∑′

k=−N/2


k

) 12π∆t

N/2∑′

k=−N/2

e−ikx(¯vl+1k − ¯vlk

)∆t dxdτ

+∫ 1

0

∫ 2π

0

~2

2m

12π

N/2∑′

k=−N/2

(−k2)eikx((1− τ)vlk + τ vl+1

k

)·

12π

N/2∑′

k=−N/2


k

)∆t dxdτ

−∫ 1

0

∫ 2π

0

12π

N/2∑′

k=−N/2

eikxVk

12π

N/2∑′

m=−N/2

eimx((1− τ)vlm + τ vl+1

m

)·

12π

N/2∑′

n=−N/2

e−inx((1− τ)¯vln + τ ¯vl+1

n

)∆t dxdt

+ λl

1− 12π

N/2∑′′

k=−N/2

vlk¯vlk

226

=∫ 1

0

i~2

12π

N/2∑′′

k=−N/2

((vl+1k − vlk)((1− τ)¯vlk + τ ¯vl+1

k )− ((1− τ)vlk + τ vl+1k )(¯vl+1

k − ¯vlk)) dτ

−∫ 1

0

~2

2πk2

2π

N/2∑′′

k=−N/2

((1− τ)vlk + τ vl+1k )((1− τ)¯vlk + τ ¯vl+1

k )

∆t dτ

−∫ 1

0

(12π

)2 −1∑′

n=−N/2

N/2+n∑′

m=−N/2

(Vn−m((1− τ)vlm + τ vl+1

m )((1− τ)¯vln + τ ¯vl+1n )

)

+N/2∑′

n=0

N/2∑′

m=n−N/2

(Vn−m((1− τ)vlm + τ vl+1

m )((1− τ)¯vln + τ ¯vl+1n )

)∆t dτ

+ λl

1− 12π

N/2∑′′

k=−N/2

vlk¯vlk

=i~4π

N/2∑′′

k=−N/2

[vl+1k

¯vlk − vlk¯vl+1k

]− ~2k2∆t

24π2

N/2∑′′

k=−N/2

[vlk(2¯vlk + ¯vl+1

k ) + vl+1k (¯vlk + 2¯vl+1

k )]

− ∆t24π2

−1∑′

n=−N/2

N/2+n∑′

m=−N/2

+N/2∑′

n=0

N/2∑′

m=n−N/2

Vn−m

[vlm(2¯vln + ¯vl+1

n ) + vl+1m (¯vln + 2¯vl+1

n )]

+ λl

1− 12π

N/2∑′′

k=−N/2

vlk¯vlk

,

where we used the fact that ∫ 2π

0

eikxdx = 2πδi0 ,

for k ∈ Z, and we define∑′ as a weighted sum where the terms with indices ±N/2 are weighted

by 1/2, and∑′′ as a weighted sum where the terms with indices ±N/2 are weighted by 1/4. We

should note that using the same approach, it would be possible to exactly evaluate the action integral

for the class of tensor product shape functions with a discrete Fourier expansion in space, and a

polynomial expansion in time. In particular, a similar approach is valid in exactly evaluating the

action integral when we use shape functions that are spectral in both space and time.

Discrete Euler–Lagrange Equations. We are now in a position to compute the discrete Euler–

Lagrange equations associated with the Schrodinger equation when using a tensor product of a

discrete Fourier expansion in space, and a linear interpolation in time.

227

The discrete variational equations are given by

0 =i~4π[¯vl−1j − ¯vl+1

j

]− ~2k2∆t

24π2

[¯vl−1j + 4¯vlj + ¯vl+1

j

]− ∆t

24π2

N/2+j∑′

n=−N/2

Vn−j[¯vl−1n + 4¯vln + ¯vl+1

n

]− λl

2π¯vlj , for j = −N/2 + 1, . . . ,−1,

0 =i~4π[vl+1j − vl−1

j

]− ~2k2∆t

24π2

[vl−1j + 4vlj + vl+1

j

]− ∆t

24π2

N/2+j∑′

n=−N/2

Vj−n[vl−1n + 4vln + vl+1

n

]− λl

2πvlj , for j = −N/2 + 1, . . . ,−1,

0 =i~4π[¯vl−1j − ¯vl+1

j

]− ~2k2∆t

24π2

[¯vl−1j + 4¯vlj + ¯vl+1

j

]− ∆t

24π2

N/2∑′

n=j−N/2

Vn−j[¯vl−1n + 4¯vln + ¯vl+1

n

]− λl

2π¯vlj , for j = 0, . . . , N/2− 1,

0 =i~4π[vl+1j − vl−1

j

]− ~2k2∆t

24π2

[vl−1j + 4vlj + vl+1

j

]− ∆t

24π2

N/2∑′

n=j−N/2

Vj−n[vl−1n + 4vln + vl+1

n

]− λl

2πvlj , for j = 0, . . . , N/2− 1,

0 =i~

16π

[¯vl−1N/2 − ¯vl+1

N/2

]− ~2k2∆t

96π2

[¯vl−1N/2 + 4¯vlN/2 + ¯vl+1

N/2

]− ∆t

48π2

N/2∑′

n=0

Vn−N/2[¯vl−1n + 4¯vln + ¯vl+1

n

]− λl

2π¯vlN/2 ,

0 =i~

16π

[vl+1N/2 − vl−1

N/2

]− ~2k2∆t

96π2

[vl−1N/2 + 4vlN/2 + vl+1

N/2

]− ∆t

48π2

N/2∑′

n=0

VN/2−n[vl−1n + 4vln + vl+1

n

]− λl

2πvlN/2 ,

1 =12π

N/2∑′′

k=−N/2

vlk¯vlk ,

0 = vl−N/2 − vlN/2 ,

0 = ¯vl−N/2 − ¯vlN/2 .

This system of (2N + 3)-equations, allow us to solve for vl+1k , ¯vl+1

k N/2k=−N/2 and λl from initial

data, vl−1k , ¯vl−1

k N/2k=−N/2 and vlk, ¯vlkN/2k=−N/2. As such, this system of equations are an exam-

ple of a spectral in space, second-order in time, pseudospectral variational integrator for the

time-dependent Schrodinger equation. The expressions for the variational integrator for the time-

228

independent Schrodinger equation, which has spectral accuracy in space, are given by

~2k2 ¯vj = −N/2+j∑′

n=−N/2

Vn−j ¯vn − λ¯vj , for j = −N/2 + 1, . . . ,−1,

~2k2vj = −N/2+j∑′

n=−N/2

Vj−nvn − λvj , for j = −N/2 + 1, . . . ,−1,

~2k2 ¯vj = −N/2∑′

n=j−N/2

Vn−j ¯vn − λ¯vj , for j = 0, . . . , N/2− 1,

~2k2vj = −N/2∑′

n=j−N/2

Vj−nvn − λvj , for j = 0, . . . , N/2− 1,

~2k2

2¯vN/2 = −

N/2∑′

n=0

Vn−N/2 ¯vn − λ¯vN/2 ,

~2k2

2vN/2 = −

N/2∑′

n=0

VN/2−nvn − λvN/2,

1 =12π

N/2∑′′

k=−N/2

vk ¯vk ,

vl−N/2 = vlN/2 ,

¯vl−N/2 = ¯vlN/2 .

As mentioned previously, it is possible generalize this approach to construct a fully spectral varia-

tional integrator in space-time, using Chebyshev polynomials to interpolate in time the coefficients

of the discrete Fourier expansion used in the spatial interpolation. The computational cost of imple-

menting such a scheme would be significantly higher, since this would require all the spatio-temporal

degrees of freedom to be solved for simultaneously.


We have introduced the notion of a generalized Galerkin variational integrator, which is based on the

idea of appropriately choosing a finite-dimensional approximation of the section of the configuration

bundle, and approximating the action integral by a numerical quadrature scheme.

In contrast to standard variational methods, that are typically formulated in terms of interpo-

latory schemes parameterized by values of field variables at nodal and internal points, generalized

Galerkin methods utilize function spaces that can be generated by arbitrary degrees of freedom.

This allows the introduction of Lie group methods, and their symmetry reduction using discrete

229

Euler–Poincare reduction, as well as multiscale, and pseudospectral methods. Nonlinear approxi-

mation spaces allow the construction of spatio-temporally adaptive methods, which are better able

to resolve shocks and other kinds of localized discontinuities in the solution.

It would be interesting to compare the performance of pseudospectral variational integrators

with traditional pseudospectral schemes to see if any additional benefits arise from constructing

pseudospectral schemes using a variational approach. More interesting still would be the compar-

ison for fully spectral methods, since both variational and non-variational methods would achieve

spectral accuracy, and it would make a particularly compelling case for variational integrators if

their advantages persist even when compared to numerical methods with spectral accuracy.

Most mesh adaptive methods use the principle of equipartitioning the error of the numerical

scheme over the mesh elements to obtain moving mesh equations. These methods rely on a posteriori

error estimators that are related to the norm in which the accuracy of the numerical method is

measured. While adaptive variational integrators exhibit an equipartitioning principle, in the sense

that the discrete conjugate momentum associated with the horizontal variations are preserved from

element to element in each connected component of the domain, it would be interesting to carefully

explore the question of whether this can be understood as arising from error equipartitioning with

respect to a geometrically motivated error estimator.

While we have only discussed the application of multiscale variational integrators to the case

of ordinary differential equations, it would be natural to consider their generalizations to partial

differential equations, whereby the multiscale shape functions are obtained through well-resolved

solutions of the cell problem, as in the case with multiscale finite elements (see, for example, Hou

and Wu [1999]). In general, short-term simulations at the fine scale can be used to construct

appropriate shape functions to obtain generalized Galerkin variational integrators at a coarser level,

through the use of principal orthogonal decomposition and balanced truncation, for example. This

is consistent with the coarse-fine computational approach proposed in Theodoropoulos et al. [2000],

or the framework of heterogeneous multiscale methods as proposed in E and Engquist [2003].

A natural generalization would be to consider wavelet based variational integrators, as well as

schemes based on conforming, hierarchical, adaptive refinement methods (CHARMS) introduced in

Grinspun et al. [2002] and further developed in Krysl et al. [2003].

231

Chapter 6

Conclusions

In this thesis, we developed discrete Routh reduction, discrete exterior calculus, discrete connections

on principal bundles, and generalized variational integrators, which can be classified into two cate-

gories, discrete geometry, and discrete mechanics, which form the basis for computational geometric

mechanics.

Computational Geometric Mechanics

Discrete Geometry Discrete Mechanics

Discrete Exterior Calculus (DEC) Discrete Routh Reduction (DRR)

Discrete Connections on Principal Bundles (DCPB) Generalized Variational Integrators (GVI)

The new machinery that has been developed will aid in the systematic development of compu-

tational algorithms that are motivated by the techniques of geometric mechanics. Some of the links

between the material in the various chapters are summarized below.

DRR and DEC. The curvature term in the discrete Routh equations can be though of as arising

from the discrete exterior derivative applied to the connection 1-form, in the case whereby the

spatial discretization goes to the continuum limit.

DRR and DCPB. Discrete connections provide the separation of the space Q×Q into horizontal

(shape) and vertical (group) components, thereby providing the coordinates necessary to realize

a discrete theory of reduction.

DRR and GVI. Generalized variational integrators provide a framework for the construction of

G-invariant discrete Lagrangians, using G-equivariant natural charts, which are necessary to

apply discrete reduction theory.

232

DEC and DCPB. Discrete connections and discrete exterior calculus provide the necessary tools

to make sense of the discrete Levi-Civita connection, and to realize its curvature as a SO(n)-

valued discrete dual 2-form, that arises from the exterior derivative of a discrete connection

expressed as a discrete dual 1-form.

DEC and GVI. Discrete exterior calculus provides a means of discretizing the action integral, and

it can be combined with multiscale or spectral discretizations in time to yield hybrid variational

schemes that capture the spatial geometry.

DCPB and GVI. Lie group variational integrators could provide an efficient method of construct-

ing discrete connections that approximate continuous connections to a prescribed degree of

accuracy.

Unifying Application. The primary motivation for developing accurate simulations is to provide

numerical results that are reliable, and minimizes the use of arbitrary parameters in the simulation

in order to obtain a correspondence with reality. In providing a link between physical models across

scales, one is in a position to accurately predict large-scale and long-term behavior, and in so doing

close the simulation and design cycle.

The next stage of evolution for numerical computation is not simply to predict what happens

given a set of initial conditions, but rather to enable simulation driven design through the use of

adjoint sensitivity techniques. This can be applied to an optimal design problem in computational

electromagnetism, which is to modify the shape of an aircraft wing by homotopy methods so as to

minimize its radar cross section. Computational electromagnetism is an application area wherein a

large number of the techniques I have been working on converge.

Reduction provides a general framework to remove the gauge symmetries in Maxwell’s equations,

and discrete exterior calculus discretizes the equations in a geometrically exact fashion. The vari-

ational framework of discrete mechanics provides a means of deriving numerical schemes that have

good structure-preserving properties, and adaptive mesh movement allow for the resolution of shocks

while minimizing computational cost. And incorporating multiscale and numerical homogenization

techniques provide the bridge to simulating the effect of complex hybrid materials on the far field

scattered wave. More generally, reduction, adaptivity, and multiscale methods increase the efficiency

of computations, while discrete exterior calculus and discrete mechanics increase the accuracy. Only

by having accurate and efficient numerical methods can one hope to realize the promise of simulation

driven design.

233

Appendix A

Review of Homological Algebra

For the reader’s convenience, we will recall some basic definitions and results from homological

algebra, which we have reproduced from Hungerford [1974].

Definition A.1. A pair of homomorphisms,

Af

// Bg

// C ,

is said to be exact at B if

Im f = Ker g .

A finite sequence of homomorphisms,

A0f1 // A1

f2 // A2f3 // · · ·

fn−1// An−1

fn // An ,

is exact if

Im fi = Ker fi+1, for i = 1, 2, . . . , n− 1.

An infinite sequence of homomorphisms,

· · ·fi−1

// Ai−1fi // Ai

fi+1// Ai+1

fi+2// · · · ,

is exact if

Im fi = Ker fi+1, for all i ∈ Z.

Remark A.1. We record below some of the properties of exact sequences.

1. The sequence 0 // Af

// B is exact iff f is a monomorphism (one-to-one).

2. The sequence Bg

// C // 0 is exact iff g is a epimorphism (onto).

234

3. If Af

// Bg

// C is exact, then gf = 0.

4. If Af

// Bg

// C // 0 is exact, then

Coker f = B/ Im f = B/Ker g = Coim g ∼= C.

5. An exact sequence of the form 0 // Af

// Bg

// C // 0 , is called a short exact

sequence, and in particular, f is a monomorphism, and g is an epimorphism.

6. A short exact sequence is another way of presenting a submodule (A ∼= Im f) and its quotient

module (B/ Im f = B/ ker g ∼= C).

We will now consider some results for short exact sequences, such as

0 // A1

f//

ook

___ Bg

//oo

h___ A2

// 0 ,

and their splittings.

Lemma A.1 (The Short Five Lemma). Consider a commutative diagram,

0 // Af

//

α

Bg

//

β

C //

γ

0

0 // A′f ′

// B′ g′// C ′ // 0

such that each row is a short exact sequence. Then,

1. α and γ are monomorphisms, implies β is a monomorphism;

2. α and γ are epimorphisms, implies β is an epimorphism;

3. α and γ are isomorphisms, implies β is an isomorphism.

Proof. The proof involves diagram chasing and the exactness of the rows. See, for example, page

176 of Hungerford [1974].

The short five lemma allows the following theorem to be proved. This theorem can be used to

relate the various representations of a connection on a principal bundle, in both the continuous and

discrete cases.

Theorem A.2. Given a short exact sequence

0 // A1f

// Bg

// A2// 0 ,

235

the following conditions are equivalent.

1. There is a homomorphism h : A2 → B with g h = 1A2 ;

2. There is a homomorphism k : B → A1 with k f = 1A1 ;

3. The given sequence is isomorphic (with identity maps on A1 and A2) to the direct sum short

exact sequence,

0 // A1i1 // A1 ⊕A2

π2 // A2// 0 ,

and in particular, B ∼= A1 ⊕A2.

A short exact sequence that satisfies the equivalent conditions of Theorem A.2 is said to be split

or a split exact sequence. The maps in Theorem A.2 are referred to as splittings of the short

exact sequence.

Proof. We present the proof sketched on pages 177–178 of Hungerford [1974].

1 ⇒ 3. Consider the homomorphism ϕ : A1⊕A2 → B, given by (a1, a2) 7→ f(a1)+h(a2), and verify

that the diagram

0 // Ai1 //

1A1

A1 ⊕A2π2 //

ϕ

A2//

1A2

0

0 // A1f

// Bg

//oo

h____ A2

// 0

is commutative. Use the short five lemma to conclude that ϕ is an isomorphism.

2 ⇒ 3. Consider the homomorphism ψ : B → A1 ⊕ A2, given by b 7→ (k(b), g(b)), and verify that

the diagram

0 // A1

f//

ook

____ Bg

// A2// 0

0 // Ai1 //

1A1

A1 ⊕A2π2 //

ψ

A2//

1A2

0

is commutative. Use the short five lemma to conclude that ψ is an isomorphism.

3 ⇒ 1, 2. Consider the commutative diagram

0 // A1

i1 //ooπ1

___ A1 ⊕A2

π2 //ooi2

___ A2// 0

0 // A1f

//

1A1

Bg

//

ϕ

A2//

1A2

0

with exact rows, and where ϕ is an isomorphism. Let h = ϕi2 : A2 → B and k = π1ϕ−1 : B →

A1, and show using the commutativity of the diagram that kf = 1A1 and gh = 1A2 .

237

Appendix B

Geometry of the Special Euclidean Group

To allow the reader to apply the construction of the exact discrete connection using exponentials

and logarithms to problems arising in geometric control, we review some of the basic geometry of

the Special Euclidean Group in three dimensions, SE(3), which is the Lie group consisting of

isometries of R3. A more detail discussion of the geometry of SE(3), and its applications to robotics

can be found in Murray et al. [1994].

Representation of SE(3). The group SE(3) is a semidirect product of SO(3) and R3. Using

homogeneous coordinates, we can represent SE(3) as follows,

SE(3) =

R p

0 1

∈ GL(4,R)

∣∣∣∣∣R ∈ SO(3), p ∈ R3

with the action on R3 given by the usual matrix-vector product when we identify R3 with the section

R3 × 1 ⊂ R4. In particular, given

g =

R p

0 1

∈ SE(3),

and q ∈ R3, we have

g · q = Rq + p,

or as a matrix-vector product, R p

0 1

q1

=

Rq + p

1

.

238

The Lie algebra of SE(3) is given by

se(3) =

ω v

0 0

∈M4(R)

∣∣∣∣∣ ω ∈ so(3), v ∈ R3

,

where · : R3 → so(3) is given by

ω =

0 −ωz ωy

ωz 0 −ωx−ωy ωx 0

.

Exponentials and Logarithms. The exponential map, exp : se(3) → SE(3), is given by

exp

ω v

0 0

=

exp(ω) Av

0 1

,

where

A = I +1− cos ‖ω‖

‖ω‖2ω +

‖ω‖ − sin ‖ω‖‖ω‖3

ω2,

and exp(ω) is given by the Rodriguez’ formula,

exp(ω) = I +sin ‖ω‖‖ω‖

ω +1− cos ‖ω‖

‖ω‖2ω2.

The logarithm, log : SE(3) → se(3), is given by

log

R p

0 1

=

log(R) A−1p

0 0

,

where

log(R) =φ

2 sinφ(R−RT ) ≡ ω,

and φ satisfies

Tr(R) = 1− 2 cosφ, |φ| < π,

and where

A−1 = I − 12ω +

2 sin ‖ω‖ − ‖ω‖(1 + cos ‖ω‖)2‖ω‖2 sin ‖ω‖

ω2.

239

Appendix C

Analysis of Multiscale Finite Elements in One

Dimension

This appendix will analyze the discrete l∞ error for multiscale finite elements (MsFEM) in one

dimension when applied to a multiscale second-order elliptic equation with homogeneous boundary

conditions. This will serve to motivate the use of multiscale shape functions in the construction of

variational integrators, for problems with multiple temporal scales, as discussed in §5.7.

Let a(y) be a smooth, periodic function in y, with period 1. Moreover, we assume that a(y) ≥

c1 > 0 for some positive constant c1, and that f(x) is a smooth function. Let ε > 0 be a small

parameter. Consider the following second-order elliptic PDE,

∂

∂x

(a(xε

) ∂

∂xuε (x)

)= f(x), 0 < x < 1 ,

with homogeneous boundary conditions, uε(0) = 0 = uε(1).

Analytical Solution. To obtain convergence estimates, it is relevant to consider the analytical

solution of the above PDE. We have

∂

∂x

(a(xε

) ∂

∂xuε (x)

)= f (x) ,

a(xε

) ∂

∂xuε (x)− a (0)

∂

∂xuε (0) =

∫ x

0

f (s) ds .

Denoting a(0) ∂∂xuε(0) by c, we obtain

∂

∂xuε (x) =

∫ x0f (s) ds− c

a(xε

) ,

uε (x) =∫ x

0

∫ y0f (s) ds− c

a(yε

) dy .

240

We impose the boundary condition uε(1) = 0, which yields

0 = uε(1)

=∫ 1

0

∫ y0f(s)dsa(yε

) dy − c

∫ 1

0

dy

a(yε

) ,c =

∫ 1

0

R y0 f(s)ds

a( yε ) dy∫ 1

0dy

a( yε )

.

Hence,

uε (x) =∫ x

0

∫ y0f (s) dsa(yε

) dy −

∫ 1

0

R y0 f(s)ds

a( yε ) dy∫ 1

0dy

a( yε )

∫ x

0

dy

a(yε

)=∫ x

0

F (y)a(yε

)dy −∫ 1

0F (y)

a( yε )dy∫ 1

0dy

a( yε )

∫ x

0

dy

a(yε

) .Analytical Expressions for the MsFEM Shape Functions. The MsFEM shape functions

can be obtained analytically as follows,

∂xa(xε

)∂xϕ

εi = 0 ,

a(xε

)∂xϕ

εi = c1 ,

∂xϕεi =

c1

a(xε

) .For x ∈ [xi−1, xi], we have

ϕεi(x) = c1

∫ x

xi−1

ds

a(sε

) ,ϕεi(xi) = c1

∫ xi

xi−1

ds

a(sε

) = 1 ,

c1 =1∫ xi

xi−1

ds

a( sε ),

ϕεi (x) =

∫ xxi−1

ds

a( sε )∫ xi

xi−1

ds

a( sε ).

For x ∈ [xi, xi+1], we have

ϕεi(xi+1)− ϕεi(x) = c1

∫ xi+1

x

ds

a(sε

) ,

241

0− 1 = ϕεi (xi+1)− ϕεi (xi)

= c1

∫ xi+1

xi

ds

a(sε

) ,c1 = − 1∫ xi+1

xi

ds

a( sε ),

ϕεi (x) =

∫ xi+1

xds

a( sε )∫ xi+1

xi

ds

a( sε ).

Then, in general,

ϕεi (x) =

[∫ xi

xi−1

ds

a( sε )

]−1 [∫ xxi−1

ds

a( sε )

], x ∈ [xi−1, xi] ;[∫ xi+1

xi

ds

a( sε )

]−1 [∫ xi+1

xds

a( sε )

], x ∈ (xi, xi+1] ;

0 , otherwise .

Discrete Error Analysis of the One-Dimensional MsFEM. Consider a uniform partition

on the interval I = [0, 1],

P : 0 = x0 < x1 < . . . < xN = 1 ,

with mesh size h = 1/N. Further, let Ii = [xi, xi+1] . We define the discrete l2 and l∞ norms as

follows,

‖f‖l2 =

(N∑i=0

|f (xi)|2)1/2

,

‖f‖l∞ = maxi=0,...,N

|f (xi)| .

In this section, we will show that the l∞ error for one-dimensional MsFEM is zero, and in particular,

the l2 error for one-dimensional MsFEM is zero as well.

The stiffness matrix is given by

Ahij = a(ϕεi , ϕ

εj

)= −

∫ 1

0

a(xε

)∇ϕεi∇ϕεjdx

=

−∫ xi+1

xi−1a(xε

)∂xϕ

εi∂xϕ

εjdx , j = i− 1, i, i+ 1 ;

0 , otherwise .

242

Since ϕεi−1 + ϕεi ≡ 1 in Ii−1 for all i, we have, ∂xϕεi−1 + ∂xϕεi ≡ 0 in Ii−1, and hence,

Ahii = −∫Ii−1

a(xε

)∂xϕ

εi∂xϕ

εidx−

∫Ii

a(xε

)∂xϕ

εi∂xϕ

εidx

= −∫Ii−1

a(xε

)∂xϕ

εi

(−∂xϕεi−1

)dx−

∫Ii

a(xε

)∂xϕ

εi

(−∂xϕεi+1

)dx

= −

(−∫Ii−1+Ii−2

a(xε

)∂xϕ

εi∂xϕ

εi−1dx−

∫Ii+Ii+1

a(xε

)∂xϕ

εi∂xϕ

εi+1dx

)= −

(Ahii−1 +Ahii+1

),

where the second to last equality is because supp(∂xϕ

εi∂xϕ

εi−1

)⊂ Ii−1.

We note further that a(·, ·) is a symmetric bilinear form, and consequently,

Ahij = Ahji.

Let us define Bhi ≡ Ahii−1, and therefore, Ahii+1 = Ahi+1,i = Bhi+1. This allows us to conclude that

(AhUh

)i= AhijU

hj

= Ahii+1Uhi+1 +AhiiU

hi +Ahii−1U

hi−1

= Ahii+1Uhi+1 −

(Ahii+1 +Ahii−1

)Uhi +Ahii−1U

hi−1

= Ahii+1

(Uhi+1 − Uhi

)−Ahii−1

(Uhi − Uhi−1

)= D+

(Ahii−1

(Uhi − Uhi−1

))= D+

(Bhi(Uhi − Uhi−1

))= D+

(Bhi D

−Uhi),

where the forward and backward difference operators D+ and D− are defined by

D+ (f (ui)) = f (ui+1)− f (ui) ,

D− (f (ui)) = f (ui)− f (ui−1) .

The MsFEM equation is given by

AhijUhj = fhi ,

where fhi =∫ 1

0f (x)ϕi (x) dx. In particular,

fhi =∫ 1

0

f (x)ϕi (x) dx

243

=

[∫ xi+1

xi

dx

a(xε

)]−1 ∫ xi+1

xi

f (x)

[∫ xi+1

x

dx

a(xε

)] dx+

[∫ xi

xi−1

dx

a(xε

)]−1 ∫ xi

xi−1

f (x)

[∫ x

xi−1

dx

a(xε

)] dx=

[∫ xi+1

xi

dx

a(xε

)]−1∫ xi+1

xi

F (x)a(xε

)dx−∫ 1

0F (x)

a( xε )dx∫ 1

0dx

a( xε )

∫ xi+1

xi

dx

a(xε

)

−

[∫ xi

xi−1

dx

a(xε

)]−1∫ xi

xi−1

F (x)a(xε

)dx−∫ 1

0F (x)

a( xε )dx∫ 1

0dx

a( xε )

∫ xi

xi−1

dx

a(xε

) ,

where we performed an integration by parts to obtain the last equality. We will further rewrite the

expression above using the difference operators.

fhi = D+

[∫ xi

xi−1

dx

a(xε

)]−1∫ xi

xi−1

F (x)a(xε

)dx−∫ 1

0F (x)

a( xε )dx∫ 1

0dx

a( xε )

∫ xi

xi−1

dx

a(xε

)

= D+

[∫ xi

xi−1

dx

a(xε

)]−2 [∫ xi

xi−1

1a(xε

)dx] (uε (xi)− uε (xi−1))

= D+

−∫ xi

xi−1

a(xε

)[∫ xi

xi−1

dx

a(xε

)]−11

a(xε

)[∫ xi

xi−1

dx

a(xε

)]−1−1a(xε

) dx

D−uε (xi)

(C.0.1)

Recall from our analytical expression for the MsFEM shape functions that

ϕεi (x) =

[∫ xi

xi−1

ds

a( sε )

]−1 [∫ xxi−1

ds

a( sε )

], x ∈ [xi−1, xi] ;[∫ xi+1

xi

ds

a( sε )

]−1 [∫ xi+1

xds

a( sε )

], x ∈ (xi, xi+1] ;

0 , otherwise ,

and hence,

∂xϕεi (x) =

[∫ xi

xi−1

ds

a( sε )

]−11

a( xε ) , x ∈ [xi−1, xi] ;[∫ xi+1

xi

ds

a( sε )

]−1−1

a( xε ) , x ∈ (xi, xi+1] ;

0 , otherwise .

We therefore recognize two of the terms in Equation C.0.1 as ∂xϕεi and ∂xϕεi−1. Consequently, we

244

have that

fhi = D+

(−

(∫ xi

xi−1

a(xε

)∂xϕ

εi∂xϕ

εi−1dx

)D−uε (xi)

)= D+

(Ahii−1D

−uε (xi))

= D+(Bhi D

−uε (xi)).

Hence,

Ahijuε (xj) = D+

(Bhi D

−uε (xi))

= fi = AhijUhj .

Since the matrix Ah is invertible, we can conclude that

Uhi = uε (xi) .

This implies that one-dimensional MsFEM is exact at the nodal points, and in particular,

‖uε − uh‖l2 = 0 ,

and

‖uε − uh‖l∞ = 0 .

245

Bibliography

R. Abraham and J. E. Marsden. Foundations of Mechanics. Addison-Wesley, second edition, 1978.

(with the assistance of Tudor Ratiu and Richard Cushman). 1, 9

R. Abraham, J. E. Marsden, and T. S. Ratiu. Manifolds, Tensor Analysis and Applications, vol-

ume 75 of Applied Mathematical Sciences. Springer-Verlag, second edition, 1988. 75, 89, 91, 94,

101, 102

D. H. Adams. R-torsion and linking numbers from simplicial abelian gauge theories. arXiv, hep-

th/9612009, 1996. 76, 77, 88

R. Almeida and P. Molino. Suites d’Atiyah et feuilletages transversalement complets. C. R. Acad.

Sci. Paris Ser. I Math., 300(1):13–15, 1985. 148

D. N. Arnold. Differential complexes and numerical stability. In Proceedings of the International

Congress of Mathematicians, Vol. I (Beijing, 2002), pages 137–157, Beijing, 2002. Higher Ed.

Press. 138

V. I. Arnold. Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Math-

ematics. Springer-Verlag, 1989. Translated from the 1974 Russian original by K. Vogtmann and

A. Weinstein. 1

M. F. Atiyah. Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc., 85:181–207,

1957. 148

M. J. Baines. Moving Finite Elements. Numerical Mathematics and Scientific Computation. Oxford

University Press, 1995. 213

M. Berry. Anticipations of the geometric phase. Phys. Today, pages 34–40, December 1990. 140

A. M. Bloch. Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics.

Springer-Verlag, 2003. 191

A. I. Bobenko, B. Lorbeer, and Y. B. Suris. Integrable discretizations of the Euler top. J. Math.

Phys., 39(12):6668–6683, 1998. 9

246

A. I. Bobenko and Y. B. Suris. Discrete Lagrangian reduction, discrete Euler–Poincare equations,

and semidirect products. Lett. Math. Phys., 49(1):79–93, 1999. 9, 72

A. Bossavit. Computational Electromagnetism. Electromagnetism. Academic Press, 1998. Varia-

tional formulations, complementarity, edge elements. 4

A. Bossavit. Generalized finite differences in computational electromagnetics. Progress in Electro-

magnetics Research, PIER, 32:45–64, 2001. 77

A. Bossavit. Applied differential geometry (a compendium). URL http://www.icm.edu.pl/

edukacja/mat/Compendium.php. (preprint), 2002a. 77

A. Bossavit. Extrusion, contraction: Their discretization via Whitney forms. (preprint), 2002b. 77,

104, 105

A. Bossavit. On “generalized finite differences”: Discretization of electromagnetic problems.

(preprint), 2002c. 76, 77

A. Cannas da Silva and A. Weinstein. Geometric Models for Noncommutative Algebras, volume 10

of Berkeley Mathematics Lecture Notes. American Mathematical Society, 1999. 129

J. R. Cardoso and F. Silva Leite. Theoretical and numerical considerations about Pade approximants

for the matrix logarithm. Linear Algebra Appl., 330(1-3):31–42, 2001. 181

E. Cartan. Geometry of Riemannian spaces. Math. Sci. Press, Brookline, CA, 1983. (translated

from French). 188

E. Cartan. Riemannian Geometry in an Orthogonal Frame. World Scientific, 2001. (translated from

French). 188

M. Castrillon-Lopez. Discrete variational problems on forms. (in preparation), 2003. 123, 135

E. Celledoni and A. Iserles. Approximating the exponential from a Lie algebra to a Lie group. Math.

Comp., 69(232):1457–1480, 2000. 181

E. Celledoni and A. Iserles. Methods for the approximation of the matrix exponential in a lie-

algebraic setting. IMA J. Num. Anal., 21(2):463–488, 2001. 181

H. Cendra, D. D. Holm, J. E. Marsden, and T. S. Ratiu. Lagrangian reduction, the Euler-Poincare

equations, and semidirect products. In Geometry of Differential Equations, volume 186 of Amer-

ican Mathematical Society Translations Series 2, pages 1–25. American Mathematical Society,

1998. 72

http://www.icm.edu.pl/edukacja/mat/Compendium.php

http://www.icm.edu.pl/edukacja/mat/Compendium.php

247

H. Cendra, J. E. Marsden, and T. S. Ratiu. Lagrangian reduction by stages. Mem. Amer. Math.

Soc., 152(722), 2001. 3, 9, 150, 168, 176, 186

D. Chang and J. E. Marsden. Geometric derivation of the Delaunay variables and geometric phases.

Nonlinearity, 16(2):1257–1275, 2003. 57

Z. Chen and T. Y. Hou. A mixed multiscale finite element method for elliptic problems with

oscillating coefficients. Math. Comp., 72(242):541–576 (electronic), 2003. 197, 215

J. Cortes. Geometric, Control and Numerical Aspects of Nonholonomic Systems, volume 1793 of

Lecture Notes in Mathematics. Springer-Verlag, 2002. 191

M. Desbrun, A. N. Hirani, M. Leok, and J. E. Marsden. Discrete exterior calculus. (in preparation),

2003a. 3, 72

M. Desbrun, A. N. Hirani, M. Leok, and J. E. Marsden. Discrete Poincare lemma. Appl. Numer.

Math., 2003b. (submitted). 108

R. A. DeVore. Nonlinear approximation. In Acta Numerica, volume 7, pages 51–150. Cambridge

University Press, 1998. 197, 213

A. A. Dezin. Multidimensional Analysis and Discrete Models. CRC Press, 1995. 77, 88

W. E and B. Engquist. The heterogeneous multiscale methods. Commun. Math. Sci., 1(1):87–132,

2003. 229

M. Eastwood. A complex from linear elasticity. In The Proceedings of the 19th Winter School

“Geometry and Physics” (Srnı, 1999), number 63 in Rend. Circ. Mat. Palermo (2) Suppl., pages

23–29, 2000. 138

Y. R. Efendiev, T. Y. Hou, and X. -H. Wu. Convergence of a nonconforming multiscale finite element

method. SIAM J. Numer. Anal., 37(3):888–910 (electronic), 2000. 197, 215

R. Forman. Discrete Morse theory and the cohomology ring. Trans. Amer. Math. Soc., 354(12):

5063–5085 (electronic), 2002. 76

P. Goldreich and A. Toomre. Some remarks on polar wandering. J. Geophys. Res., 10:2555–2567,

1969. 143

E. Grinspun, P. Krysl, and P. Schroder. CHARMS: A simple framework for adaptive simulation.

ACM Transactions on Graphics (SIGGRAPH), 21(21):281–290, July 2002. 229

P. W. Gross and P. R. Kotiuga. Data structures for geometric and topological aspects of finite

element algorithms. Progress in Electromagnetics Research, PIER, 32:151–169, 2001. 77

248

E. Hairer, C. Lubich, and G. Wanner. Geometric Numerical Integration, volume 31 of Springer

Series in Computational Mathematics. Springer-Verlag, 2002. Structure-preserving algorithms for

ordinary differential equations. 1

E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I : Nonstiff

problems, volume 8 of Springer Series in Computational Mathematics. Springer-Verlag, second

edition, 1993. 36

E. Hairer and G. Wanner. Solving Ordinary Differential Equations II : Stiff and differential-algebraic

problems, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, second

edition, 1996. 36

A. Hatcher. Algebraic Topology. Cambridge University Press, 2001. 94, 97

N. J. Higham. Evaluating Pade approximants of the matrix logarithm. SIAM J. Matrix Anal. Appl.,

22(4):1126–1135 (electronic), 2001. 181

R. Hiptmair. Canonical construction of finite elements. Math. Comp., 68(228):1325–1346, 1999. 4,

77, 88

R. Hiptmair. Discrete Hodge-operators: An algebraic perspective. Progress in Electromagnetics

Research, PIER, 32:247–269, 2001a. 77

R. Hiptmair. Higher order Whitney forms. Progress in Electromagnetics Research, PIER, 32:271–

299, 2001b. 77

R. Hiptmair. Finite elements in computational electromagnetism. In Acta Numerica, volume 11,

pages 237–339. Cambridge University Press, 2002. 77, 92

A. N. Hirani. Discrete Exterior Calculus. PhD thesis, California Institute of Technology, 2003. 94,

105

D. D. Holm, J. E. Marsden, and T. S. Ratiu. Euler–Poincare models of ideal fluids with nonlinear

dispersion. Phys. Rev. Lett., 349:4173–4177, 1998. 9, 72

T. Y. Hou and X. -H. Wu. A multiscale finite element method for PDEs with oscillatory coefficients.

In Numerical treatment of multi-scale problems (Kiel, 1997), volume 70 of Notes Numer. Fluid

Mech., pages 58–69. Vieweg, 1999. 197, 215, 229

T. W. Hungerford. Algebra, volume 73 of Graduate Texts in Mathematics. Springer-Verlag, 1974.

233, 234, 235

249

C. L. Hwang and L. T. Fan. A discrete version of Pontryagin’s maximum principle. Operations Res.,

15:139–146, 1967. 8

J. M. Hyman and M. Shashkov. Adjoint operators for the natural discretizations of the divergence,

gradient and curl on logically rectangular grids. Appl. Numer. Math., 25(4):413–442, 1997a. 4

J. M. Hyman and M. Shashkov. Natural discretizations for the divergence, gradient, and curl on

logically rectangular grids. Comput. Math. Appl., 33(4):81–104, 1997b. 4

A. Iserles. On the numerical quadrature of highly-oscillating integrals I: Fourier transforms. Tech-

nical Report 2003/NA05, DAMTP, Cambridge, 2003a. (to appear in IMA J. Num. Anal.). 220

A. Iserles. On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators. Tech-

nical Report 2003/NA09, DAMTP, Cambridge, 2003b. 220

A. Iserles. On the method of Neumann series for highly oscillatory equations. Technical Report

2004/NA02, DAMTP, Cambridge, 2004. 220

A. Iserles, H. Munthe-Kaas, S. P. Nørsett, and A. Zanna. Lie-group methods. In Acta Numerica,

volume 9, pages 215–365. Cambridge University Press, 2000. 203

A. Iserles and S. P. Nørsett. Efficient quadrature of highly oscillatory integrals using derivatives.

Technical Report 2004/NA03, DAMTP, Cambridge, 2004. 220

J. D. Jackson. Classical Electrodynamics. Wiley, third edition, 1998. 125

S. M. Jalnapurkar, M. Leok, J. E. Marsden, and M. West. Discrete Routh reduction. J. FoCM,

2003. (submitted). 2

S. M. Jalnapurkar and J. E. Marsden. Reduction of Hamilton’s variational principle. Dynam. Stabil.

Syst., 15(3):287–318, 2000. 9, 20, 22, 46, 72

S. M. Jalnapurkar and J. E. Marsden. Variational discretization of Hamiltonian systems. (in prepa-

ration), 2003. 8

B. W. Jordan and E. Polak. Theory of a class of discrete optimal control systems. J. Electron.

Control, 17:697–711, 1964. 8

J. V. Jose and E. J. Saletan. Classical Dynamics: A Contemporary Approach. Cambridge University

Press, 1998. 222

T. Kaczynski, K. Mischaikow, and M. Mrozek. Computational homology, volume 157 of Applied

Mathematical Sciences. Springer-Verlag, 2004. 76

250

C. Kane, J. E. Marsden, and M. Ortiz. Symplectic-energy-momentum preserving variational inte-

grators. J. Math. Phys., 40(7):3353–3371, 1999. 1, 194, 202, 210, 212, 213

C. Kane, J. E. Marsden, M. Ortiz, and M. West. Variational integrators and the Newmark algorithm

for conservative and dissipative mechanical systems. Int. J. Numer. Meth. Eng., 49(10):1295–1325,

2000. 1, 8, 51

S. Kobayashi and K. Nomizu. Foundations of Differential Geometry, volume 1. Wiley, 1963. 145,

148, 149

P. Krysl, A. Trivedi, and B . Zhu. Object-oriented hierarchical mesh refinement with CHARMS.

Int. J. Numer. Meth. Eng., 2003. (to appear). 229

S. Lall and M. West. Discrete variational mechanics and duality. (in preparation), 2003. 201

B. Leimkuhler and S. Reich. Simulating Hamiltonian Dynamics, volume 14 of Cambridge Monographs

on Applied and Computational Mathematics. Cambridge University Press, 2004. 1

M. Leok. A mathematical model of true polar wander. Caltech SURF Report, 1998. 143

M. Leok. Discrete Routh reduction and discrete exterior calculus. Caltech Control and Dynamical

Systems, Ph.D. Candidacy Report, 2002. 3

M. Leok. Generalized Galerkin variational integrators: Lie group, multiscale, and pseudospectral

methods. (in preparation), 2004. 3

M. Leok, J. E. Marsden, and A. Weinstein. A discrete theory of connections on principal bundles.

(in preparation), 2003. 3, 72, 86

A. Lew, J. E. Marsden, M. Ortiz, and M. West. Asynchronous variational integrators. Arch. Ration.

Mech. An., 167(2):85–146, 2003. 1, 74, 194

A. Lew, J. E. Marsden, M. Ortiz, and M. West. Variational time integrators. Int. J. Numer. Meth.

Eng., 2004. (to appear). 1, 212

K. C. H. Mackenzie. Lie algebroids and Lie pseudoalgebras. Bull. London Math. Soc., 27(2):97–147,

1995. 148

S. Maeda. Lagrangian formulation of discrete systems and concept of difference space. Math.

Japonica, 27(3):345–356, 1981. 8

E. L. Mansfield and P. E. Hydon. On a variational complex for difference equations. In The Geo-

metrical Study of Differential Equations (Washington, DC, 2000), volume 285 of Contemporary

Mathematics, pages 121–129. American Mathematical Society, 2001. 77

251

J. E. Marsden. Lectures on Mechanics, volume 174 of London Mathematical Society Lecture Note

Series. Cambridge University Press, 1992. 9, 11, 12, 13, 59, 66

J. E. Marsden. Geometric mechanics, stability, and control. In L. Sirovich, editor, Applied Mathe-

matical Sciences, volume 100, pages 265–291. Springer-Verlag, 1994. 140, 143

J. E. Marsden. Geometric foundations of motion and control. In Motion, Control and Geometry,

pages 3–19. National Academy Press, 1997. 140, 143

J. E. Marsden, R. Montgomery, and T. S. Ratiu. Reduction, symmetry, and phases in mechanics.

Mem. Amer. Math. Soc., 88(436):1–110, 1990. 13, 59, 143

J. E. Marsden, G. W. Patrick, and S. Shkoller. Multisymplectic geometry, variational integrators,

and nonlinear PDEs. Commun. Math. Phys., 199(2):351–395, 1998. 8, 193, 195

J. E. Marsden, S. Pekarsky, and S. Shkoller. Discrete Euler–Poincare and Lie–Poisson equations.

Nonlinearity, 12(6):1647–1662, 1999. 2, 9, 129, 205, 207, 209

J. E. Marsden, S. Pekarsky, and S. Shkoller. Symmetry reduction of discrete Lagrangian mechanics

on Lie groups. J. Geom. Phys., 36(1-2):140–151, 2000a. 2, 9, 129

J. E. Marsden, S. Pekarsky, S. Shkoller, and M. West. Variational methods, multisymplectic geometry

and continuum mechanics. J. Geom. Phys., 38(3-4):253–284, 2001. 193, 195

J. E. Marsden and T. S. Ratiu. Introduction to Mechanics and Symmetry, volume 17 of Texts in

Applied Mathematics. Springer-Verlag, second edition, 1999. 1, 17, 44, 140

J. E. Marsden, T. S. Ratiu, and J. Scheurle. Reduction theory and the Lagrange–Routh equations.

J. Math. Phys., 41(6):3379–3429, 2000b. 9, 11, 13, 22, 46, 59, 72

J. E. Marsden and J. Scheurle. Lagrangian reduction and the double spherical pendulum. Z. Angew.

Math. Phys., 44(1):17–43, 1993a. 9, 46

J. E. Marsden and J. Scheurle. The reduced Euler-Lagrange equations. In Dynamics and Control of

Mechanical Systems (Waterloo, ON, 1992), volume 1 of Fields Institute Communications, pages

139–164. American Mathematical Society, 1993b. 9, 46

J. E. Marsden and J. M. Wendlandt. Mechanical systems with symmetry, variational principles

and integration algorithms. In M. Alber, B. Hu, and J. Rosenthal, editors, Current and Future

Directions in Applied Mathematics, pages 219–261. Birkhauser, 1997. 8

J. E. Marsden and M. West. Discrete mechanics and variational integrators. In Acta Numerica,

volume 10, pages 317–514. Cambridge University Press, 2001. 1, 8, 14, 37, 38, 194, 203

252

C. Mattiussi. An analysis of finite volume, finite element, and finite difference methods using some

concepts from algebraic topology. J. Comput. Phys., 133(2):289–309, 1997. 77

C. Mattiussi. The finite volume, finite difference, and finite element methods as numerical methods

for physical field problems. Adv. Imag. Elect. Phys., 113:1–146, 2000. 74, 77

R. McLachlan and M. Perlmutter. Integrators for nonholonomic mechanical systems. (preprint),

2003. 191

M. Meyer, M. Desbrun, P. Schroder, and A. H. Barr. Discrete differential-geometry operators for

triangulated 2-manifolds. VisMath, 2002. 92, 103

R. Montgomery. How much does a rigid body rotate? A Berry’s phase from the eighteenth century.

Amer. J. Phys., 59:394–398, 1991. 143

B. Moritz. Vector Difference Calculus. PhD thesis, University of North Dakota, 2000. 77

B. Moritz and W. A. Schwalm. Triangle lattice green functions for vector fields. J. Phys. A., 34(3):

589–602, 2001. 77

J. Moser and A. P. Veselov. Discrete versions of some classical integrable systems and factorization

of matrix polynomials. Commun. Math. Phys., 139(2):217–243, 1991. 8

S. Muller and M. Ortiz. On the Γ-convergence of discrete dynamics and variational integrators. J.

Nonlinear Sci., 2004. (to appear). 1

J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, 1984. 78, 87, 88

R. M. Murray, Z. Li, and S. S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC

Press, 1994. 237

R. A. Nicolaides and D. -Q. Wang. Convergence analysis of a covolume scheme for Maxwell’s

equations in three dimensions. Math. Comp., 67(223):947–963, 1998. 77

J. Nocedal and S. J. Wright. Numerical Optimization. Springer Series in Operations Research.

Springer-Verlag, 1999. 197

S. P. Novikov. Discrete connections on the triangulated manifolds and difference linear equations.

arXiv, math-ph/0303035, 2003. 188

M. Oliver, M. West, and C. Wulff. Approximate momentum conservation for spatial semidiscretiza-

tions of nonlinear wave equations. Numer. Math., 2004. (to appear). 213

253

P. J. Olver. Geometric foundations of numerical algorithms and symmetry. Appl. Algebra Engrg.

Comm. Comput., 11(5):417–436, 2001. Special issue “Computational geometry for differential

equations”. 177

J. -P. Ortega and T. S. Ratiu. Hamiltonian Singular Reduction. Progress in Mathematics. Birkhauser,

2001. (to appear). 65

G. W. Patrick. Two Axially Symmetric Coupled Rigid Bodies: Relative Equilibria, Stability Bifur-

cations, and a Momentum Preserving Symplectic Integrator. PhD thesis, University of California

at Berkeley, 1991. 71

S. Pekarsky and M. West. Discrete diffeomorphism groupoids and circulation conserving fluid inte-

grators. (in preparation), 2003. 131

J. E. Prussing and B. A. Conway. Orbital Mechanics. Oxford University Press, 1993. 57

E. J. Routh. Stability of a Given State of Motion. Macmillian, 1877. Reprinted in Stability of

Motion, A.T. Fuller (ed.), Halsted Press, 1975. 9, 43

E. J. Routh. Advanced Rigid Dynamics. Macmillian, 1884. 9

A. Sanyal, J. Shen, and N. H. McClamroch. Variational integrators for mechanical systems with

cyclic generalized coordinates. (preprint), 2003. 45

J. M. Sanz-Serna and M. P. Calvo. Numerical Hamiltonian Problems, volume 7 of Applied Mathe-

matics and Mathematical Computation. Chapman & Hall, 1994. 1

J. R. Schewchuck. What is a good linear finite element? Interpolation, conditioning, anisotropy

and quality measures. URL http://www.cs.berkeley.edu/~jrs/papers/elemj.ps. (preprint),

2002. 79

W. A. Schwalm, B. Moritz, M. Giona, and M. K. Schwalm. Vector difference calculus for physical

lattice models. Phys. Rev. E., 59(1, part B):1217–1233, 1999. 77

S. Sen, S. Sen, J. C. Sexton, and D. H. Adams. Geometric discretization scheme applied to the

abelian Chern–Simons theory. Phys. Rev. E, 61(3):3174–3185, 2000. 76, 77, 88, 91

A. Shapere and F. Wilczek. Geometric Phases in Physics, volume 5 of Advanced Series in Mathe-

matical Physics. World Scientific, 1989. 140

M. Shashkov. Conservative finite-difference methods on general grids. Symbolic and Numeric Com-

putation Series. CRC Press, 1996. 4

http://www.cs.berkeley.edu/~jrs/papers/elemj.ps

254

N. Steenrod. The Topology of Fibre Bundles. Princeton University Press, 1951. 145

Y. Suris. Hamiltonian methods of Runge–Kutta type and their variational interpretation. Mat.

Model., 2(4):78–87, 1990. 36, 37

F. L. Teixeira. Geometric aspects of the simplicial discretization of Maxwell’s equations. Progress

in Electromagnetics Research, PIER, 32:171–188, 2001. 77

K. Theodoropoulos, Y. -H. Qian, and I. G. Kevrekidis. “Coarse” stability and bifurcation analysis

using timesteppers: a reaction diffusion example. Proc. Natl. Acad. Sci., 97(18):9840–9843, 2000.

229

P. Thoutireddy and M. Ortiz. A variational r-adaptation and shape-optimization method for finite-

deformation elasticity. Int. J. Numer. Meth. Eng., 2003. (to appear). 213

Y. Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun. Discrete multiscale vector field decompo-

sition. ACM Transactions on Graphics (SIGGRAPH), July 2003. 75, 101

E. Tonti. Finite formulation of electromagnetic field. IEEE Trans. Mag., 38:333–336, 2002. 77

L. N. Trefethen. Spectral methods in MATLAB. Software, Environments, and Tools. Society for

Industrial and Applied Mathematics (SIAM), 2000. 222

V. Vedral. Geometric phases and topological quantum computation. Int. J. Quantum Inform., 1

(1):1–23, 2003. 140

A. P. Veselov. Integrable discrete-time systems and difference operators. Funct. Anal. Appl., 22(2):

83–93, 1988. 8

A. P. Veselov. Integrable Lagrangian correspondences and the factorization of matrix polynomials.

Funct. Anal. Appl., 25(2):112–122, 1991. 8

G. Walsh and S. Sastry. On reorienting linked rigid bodies using internal motions. IEEE Trans.

Robotic Autom., 11:139–146, 1993. 143

A. Weinstein. Lagrangian mechanics and groupoids. In Mechanics Day (Waterloo, ON, 1992),

volume 7 of Fields Institute Communications, pages 207–231. American Mathematical Society,

1996. 129

A. Weinstein. Groupoids: unifying internal and external symmetry. A tour through some examples.

In Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), volume 282 of Contem-

porary Mathematics, pages 1–19. American Mathematical Society, 2001. 129

255

J. M. Wendlandt and J. E. Marsden. Mechanical integrators derived from a discrete variational

principle. Physica D, 106(3-4):223–246, 1997. 8

H. Whitney. Geometric Integration Theory. Princeton University Press, 1957. 76

J. Wisdom, S. J. Peale, and F. Mignard. The chaotic rotation of Hyperion. Icarus, 58(2):137–152,

1984. 71

Z. J. Wood. Computational Topology Algorithms for Discrete 2-Manifolds. PhD thesis, California

Institute of Technology, 2003. 76

A. Zanna and H. Z. Munthe-Kaas. Generalized polar decompositions for the approximation of the

matrix exponential. SIAM J. Matrix Anal. Appl., 23(3):840–862 (electronic), 2001/02. 181

257

Index

Symbols

FL, see Legendre transform

FLd, see Legendre transform, discrete

J , see momentum map

JL, see momentum map, Lagrangian

Jd, see momentum map, discrete

Ld, see Lagrangian, discrete

Rµ, see Routhian

Xd(K), see vector field, primal

Xd(?K), see vector field, dual

∆, see Laplace–Beltrami

div, see divergence

Ωkd(K), see form, primal

Ωkd(?K), see form, dual

∗, see Hodge star

δ, see codifferential

[, see flat

κ, see causality sign

], see sharp

?, see circumcentric, duality operator

∧, see wedge product

d, see exterior derivative

iX , see contraction

A

action

discrete, 14

Atiyah sequence, 148

discrete, see discrete Atiyah sequence

augmentation

example, 114, 117

one-ring cone, 114

B

boundary, 89

dual, 91

example, 90

bundle

G-bundle, 145

adjoint, 148

base space, 145

bundle space, 145

fiber, 145

fiber bundle, 145

principal bundle, 145

projection, 145

structure group, 145

C

causality sign, 127

example, 127

cell

complex, 84

example, 84, 85

chain, 87

complex, 90

example, 87

change of variables, 135

258

circumcenter, 79

circumcentric

subdivision, 79

dual, 79

duality operator, 79

coboundary, 90

cochain, 88

complex, 90

cocone, 109

codifferential, 92

complex

cell, see cell, complex

chain, see chain, complex

cochain, see cochain, complex

prismal, 126

simplicial, see simplicial, complex

sub, see subcomplex

cone, 109

example, 112, 115

generalized, 113

geometric, 109

logical, 112

connection, 147

1-form, 148

discrete, see discrete connection

from discrete connection, 176

mechanical, 11

principal, 10

to discrete connection, 181

constraints

structured, 74

contraction

algebraic, 105

extrusion, 105

curvature

Levi-Civita connection, see connection, Levi-

Civita, curvature

D

diffeomorphism, 131

groupoid, 133

interpolatory methods, 133

non-degenerate, 132

one-parameter family, 132

discrete Atiyah sequence, 154

splitting, 165, 167

from discrete connection 1-form, 166

from discrete horizontal lift, 167

to discrete connection 1-form, 166

to discrete horizontal lift, 168

discrete connection

1-form, see discrete connection 1-form

computation, 178

derived geometric structures, 172

exact, 178

extended groupoid composition, 173

from continuous connection, 181



geometric control, 186

higher-order tangent bundle, 176

horizontal component, 171

isomorphism, 168

Lagrangian reduction, 185

Levi-Civita, 188

curvature, 189

mechanical, 161, 179

from discrete Lagrangian, 182

order of approximation, 180

relating the representations, 156

259

to continuous connection, 176



vertical component, 171

discrete connection 1-form



from splitting of Atiyah sequence, 166

local representation, 160

properties, 157



to splitting of Atiyah sequence, 166

discrete generator, 152

discrete horizontal lift, 161



from splitting of Atiyah sequence, 168



to splitting of Atiyah sequence, 167

discrete mechanics, 14, 121, 194

groupoid, 129

with symmetry, 17

discrete Riemannian manifold, 188

divergence, 101

dual cell

circumcentric, see circumcentric, dual

orientation, see orientation, dual cell

dynamic problems, 129

E

Euler–Lagrange

discrete operator, 185

equations, 9

discrete, 15

reduced

discrete, 186

exterior derivative, 90

extrusion, 104

contraction, see contraction, extrusion

example, 104

Lie derivative, see Lie derivative, extru-

sion

F

face, 78

fiber product, 146

Filon quadrature, 220

flat, 93

flow, 104

example, 107

form

dual, 91

primal, 88

G

generating function, 16

geometric phase

rigid-body, 143

groupoid

composition, 130

extended, see discrete connection, extended

groupoid composition

diffeomorphism, see groupoid, diffeomor-

phism

discrete mechanics, see discrete mechan-

ics, groupoid

inverse, 131

source, 130

target, 130

260

visualizing, 131

H

Hamiltonian, 10

vector field, 10

Harmonic functions, 123

harmonic functions

action functional, 123

Euler–Lagrange, 123

highly-oscillatory integral, 220

Hodge star, 91

Lorentizan space, 127

holonomy, 140

horizontal

component, 153

discrete, see discrete connection, horizon-

tal component

directions, 147

lift, 149

discrete, see discrete horizontal lift

space, 11

discrete, 153

I

inner product, 122

integrator, 17

L

Lagrange 1-forms

discrete, 15

Lagrange 2-form

discrete, 15

Lagrange map

discrete, 15

push-forward, 16

Lagrange–Poincare

discrete operator, 185

Lagrangian, 9

discrete, 14, 194

approximate, 17

exact, 16, 194

group-regular, 20

polynomials and quadrature, 37

symplectic Runge–Kutta, 36

double spherical pendulum, 64

satellite, 57

Laplace–Beltrami, 102

Laplace–deRham, 103

lattice theory, 74

Legendre transform, 9

discrete, 16

reduced Routh, 13

Lie derivative

algebraic, 108

extrusion, 107

M

manifolds

flat, 89

non-flat, 85, 89, 137

Maxwell equations, 124



metric

local, 86, 190

momentum map, 10

discrete, 17

Lagrangian, 10

momentum shift, 12, 13

multigrid, 136

multiscale

261

shape function, 215, 218

variational integrator, see variational in-

tegrator, multiscale

multisymplectic

configuration bundle, 195

first jet bundle, 195

first jet extension, 195

geometry, 195

variational integrator, see variational in-

tegrator, multisymplectic

N

natural charts, 205

Noether’s theorem, 10

discrete, 18

norm, 123

Lorentzian, 127

O

orientation

dual cell, 81

dual of dual, 81

example, 81, 82

P

pairing

natural, 89

phase space

discrete, 14

Poincare lemma, 108

counterexample, 120

polytope, 78

pull-back

form, 135

vector field, 134

push-forward

form, 135

vector field, 134

R

reduction

contangent bundle, 42

continuous, 9

cotangent bundle, 12, 42

discrete, 14, 22

Euler–Poincare, 207

Hamiltonian, 12

Lagrange–Poincare, 185

Lagrangian, 11

reconstruction, 13

discrete, 19

relating discrete and continuous, 34

relating Lagrangian and Hamiltonian, 12

Routh, 11, 42

symplectic Runge–Kutta, see symplectic

Runge–Kutta, reduction

remeshing, 136

Routh equations, 12

discrete, 26

computational considerations, 69

constrained, 46, 47, 56

forced, 46, 51, 52, 56

preservation of symplectic form, 27


satellite, 57

Routhian, 12


satellite, 59

S

Schrodinger equation, 222


262

discrete action, 225


shape space, 10

sharp, 93

simplex, 78

example, 84, 85

examples, 78

simplicial

complex, 78

triangulation, 78

skeleton, 78

star-shaped

generalized, 114

logically, 112

trivially, 109

Stokes’ theorem

generalized, 90

subcomplex, 78

support volume, 83

example, 84, 85

symplectic Runge–Kutta, 35

reduction, 38, 42

symplectic structure

canonical, 10

reduced, 12

discrete, 27

T

trajectory

discrete, 14

V

variational integrator

discrete action, 202

Euler–Poincare, 207

function space, 197

Galerkin, 203

generalized, 197

higher-order, 198

Lie group, 202

multiscale, 215

planar pendulum with a stiff spring, 216

multisymplectic, 196, 199

numerical quadrature, 197

pseudospectral, 222

Schrodinger equation, 223

spatio-temporally adaptive, 212

symplectic-energy-momentum, 200

variational principle

discrete, 194

Hamilton’s, 11

Lagrange–d’Alembert, 51

reduced, 11

variational problems, 74

harmonic functions, see harmonic functions

Maxwell equations, see Maxwell equations

Schrodinger equation, see Schrodinger equa-

tion

vector field

dual, 92

primal, 92

velocity

material, 132

spatial, 133

vertex, 78

vertical

component, 154

discrete, see discrete connection, vertical

component

directions, 147

space, 11

263

discrete, 152

W

wedge product, 94

dual-dual, 95

anti-commutative, 95

associativity, 98

convergence, 100

example, 94

Leibniz rule, 97

natural, 135

naturality, 135

primal-dual, 122

primal-primal, 94

Whitney sum, 146

265

Vita

Melvin Leok will join the mathematics department of the University of Michigan, Ann Arbor, in

September 2004, as a T.H. Hildebrandt Research Assistant Professor. He received his B.S. with hon-

ors and M.S. in Mathematics in 2000, and his Ph.D. in Control and Dynamical Systems with a minor

in Applied and Computational Mathematics under the direction of Jerrold Marsden in 2004, all from

the California Institute of Technology. His primary research interests are in computational geomet-

ric mechanics, discrete geometry, and structure-preserving numerical schemes, and particularly how

these subjects relate to systems with symmetry and multiscale systems. He was the recipient of the

SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis, both

in 2003, for his work on Foundations of Computational Geometric Mechanics. While a doctoral

student at Caltech, he held a Poincare Fellowship (2000–2004), a Josephine de Karman Fellowship

(2003–2004), an International Fellowship from the Agency for Science, Technology, and Research

(2002–2004), a Tau Beta Pi Fellowship (2000–2001), and a Tan Kah Kee Foundation Postgradu-

ate Scholarship (2000). As a Caltech undergraduate, he received the Loke Cheng-Kim Foundation

Scholarship (1996–2000), the Carnation Scholarship (1998–2000), the Herbert J. Ryser Scholarship

(1999), the E.T. Bell Undergraduate Mathematics Research Prize (1999), and the Jack E. Froehlich

Memorial Award (1999).

foundations of computational geometric...

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